Ternary quadratic form theta series as Hecke eigenforms and class number one

Question

At Simple comparison of positive ternary quadratic form representation counts Jeremy answered:

"The reason is that the theta series for the sums of three squares form is an eigenfunction for all the Hecke operators (which is encoded by the relations $R(np^{2}) = (p+1 - \left(\frac{-n}{p}\right)) R(n) - p R(n/p^{2})$ and $R(4n) = R(n)$), while in general, the theta series for other ternary forms will not be. It should be easy to show that if $f(x,y,z)$ is such that its theta series is a Hecke eigenform, then $f(x,y,z)$ must be regular."

I did some experiments last night, and very comprehensive experiments just now, and I have a little conjecture...

Is it the case that Jeremy's condition implies class number one? I wrote a little program, all it does is test, for a form and a prime $p<15,$ and the first $20$ numbers $n$ prime to $p$ that are represented, do we always have $$ R(n p^2 ) \geq p R(n)? $$ The result: yes for the 794 class number one positive ternaries, no for the other 119 regular (or probably regular) forms, and no for evidently irregular forms.

If this is all true, do we always get ( with primes for which the form is isotropic in $\mathbb Q_p$) relations of the sort $$R(np^{2}) = (p+1 - \left(\frac{-\Delta n}{p}\right)) R(n) - p R(n/p^{2})$$ where, with $$ f(x,y,z) = a x^2 + b y^2 + c z^2 + r y z + s z x + t x y $$ we have Lehman's discriminant $$ \Delta = 4abc+rst - ar^2 - b s^2 - c t^2. $$

This is so cool.

Saturday, output showing the 119 regular (or probably regular) forms of class number larger than one fail my simplified test:

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  counted   913  regular  

==================================

       11 :     1    1    3    0    1    0
 count is 3 ; n =    5   12 ;    20   16   p =    2   ; R(n p^2 ) - p R(n) =  -8

       15 :     1    2    2    1    0    0
 count is 5 ; n =    9   22 ;    36   42   p =    2   ; R(n p^2 ) - p R(n) =  -2

       17 :     1    2    3    2    1    1
 count is 8 ; n =   15   20 ;    60   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

       21 :     1    2    3    0    0    1
 count is 6 ; n =   11   20 ;    44   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

       24 :     1    3    3    3    1    1
 count is 6 ; n =    9   22 ;   225  102   p =    5   ; R(n p^2 ) - p R(n) =  -8

       27 :     1    1    7    0    1    0
 count is 1 ; n =    1    4 ;     4    4   p =    2   ; R(n p^2 ) - p R(n) =  -4

       27 :     1    1    9    0    0    1
 count is 1 ; n =    1    6 ;     4    6   p =    2   ; R(n p^2 ) - p R(n) =  -6

       27 :     1    2    4    1    0    1
 count is 4 ; n =    7   10 ;   175   38   p =    5   ; R(n p^2 ) - p R(n) =  -12

       27 :     1    3    3    3    0    0
 count is 8 ; n =   25   38 ;   100   50   p =    2   ; R(n p^2 ) - p R(n) =  -26

       32 :     1    3    3    1    0    1
 count is 12 ; n =   25   22 ;   225   62   p =    3   ; R(n p^2 ) - p R(n) =  -4

       44 :     1    3    4    0    0    1
 count is 16 ; n =   31   36 ;   279   96   p =    3   ; R(n p^2 ) - p R(n) =  -12

       45 :     1    3    4    0    1    0
 count is 10 ; n =   27   22 ;   108   42   p =    2   ; R(n p^2 ) - p R(n) =  -2

       48 :     1    3    5    3    1    0
 count is 10 ; n =   17   20 ;   425   84   p =    5   ; R(n p^2 ) - p R(n) =  -16

       50 :     1    2    7    2    1    0
 count is 7 ; n =   13   16 ;   117   40   p =    3   ; R(n p^2 ) - p R(n) =  -8

       56 :     1    3    5    1    1    0
 count is 5 ; n =    9   10 ;   225   46   p =    5   ; R(n p^2 ) - p R(n) =  -4

       63 :     1    3    6    3    0    0
 count is 9 ; n =   25   22 ;   100   42   p =    2   ; R(n p^2 ) - p R(n) =  -2

       64 :     1    1   16    0    0    0
 count is 1 ; n =    1    4 ;     9    4   p =    3   ; R(n p^2 ) - p R(n) =  -8

       72 :     1    3    7    2    1    1
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

       72 :     1    3    7    3    1    0
 count is 7 ; n =   13   20 ;   325   84   p =    5   ; R(n p^2 ) - p R(n) =  -16

       75 :     1    4    5    0    0    1
 count is 9 ; n =   29   20 ;   116   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

       80 :     1    3    7    1    1    0
 count is 4 ; n =    9   10 ;   441   66   p =    7   ; R(n p^2 ) - p R(n) =  -4

       81 :     1    3    7    0    1    0
 count is 6 ; n =   12   14 ;   300   50   p =    5   ; R(n p^2 ) - p R(n) =  -20

      100 :     1    2   13    2    0    0
 count is 9 ; n =   17   16 ;   153   40   p =    3   ; R(n p^2 ) - p R(n) =  -8

      108 :     1    1   36    0    0    1
 count is 1 ; n =    1    6 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -24

      108 :     1    3   10    3    1    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

      108 :     1    4    7    0    1    0
 count is 7 ; n =   13   16 ;   325   32   p =    5   ; R(n p^2 ) - p R(n) =  -48

      108 :     1    5    7    5    1    1
 count is 3 ; n =    7   10 ;   175   38   p =    5   ; R(n p^2 ) - p R(n) =  -12

      120 :     1    3   11    3    1    0
 count is 9 ; n =   17   16 ;   833   80   p =    7   ; R(n p^2 ) - p R(n) =  -32

      121 :     1    3   11    0    0    1
 count is 1 ; n =    1    2 ;     4    2   p =    2   ; R(n p^2 ) - p R(n) =  -2

      135 :     1    3   12    3    0    0
 count is 5 ; n =   12   10 ;   588   62   p =    7   ; R(n p^2 ) - p R(n) =  -8

      135 :     2    2    9    0    0    1
 count is 5 ; n =   17   20 ;    68   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      144 :     1    3   13    3    1    0
 count is 9 ; n =   19   20 ;   475   84   p =    5   ; R(n p^2 ) - p R(n) =  -16

      147 :     1    2   21    0    0    1
 count is 8 ; n =   29   20 ;   116   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      189 :     2    3    8    0    1    0
 count is 4 ; n =   11   10 ;    44   18   p =    2   ; R(n p^2 ) - p R(n) =  -2

      216 :     1    3   19    3    1    0
 count is 13 ; n =   31   20 ;   775   84   p =    5   ; R(n p^2 ) - p R(n) =  -16

      216 :     3    5    5    2    3    3
 count is 13 ; n =   39   20 ;   975   84   p =    5   ; R(n p^2 ) - p R(n) =  -16

      225 :     2    2   15    0    0    1
 count is 5 ; n =   23   20 ;    92   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      240 :     1    5   13    2    1    1
 count is 1 ; n =    1    2 ;    49   10   p =    7   ; R(n p^2 ) - p R(n) =  -4

      243 :     1    7    9    0    0    1
 count is 1 ; n =    1    2 ;     4    2   p =    2   ; R(n p^2 ) - p R(n) =  -2

      243 :     2    3   11    3    1    0
 count is 9 ; n =   35   12 ;   140   20   p =    2   ; R(n p^2 ) - p R(n) =  -4

      256 :     1    2   32    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

      256 :     1    4   16    0    0    0
 count is 1 ; n =    1    2 ;     9    2   p =    3   ; R(n p^2 ) - p R(n) =  -4

      256 :     1    5   13    2    0    0
 count is 11 ; n =   29   14 ;   261   34   p =    3   ; R(n p^2 ) - p R(n) =  -8

      289 :     3    5    6    1    2    3
 count is 6 ; n =   23   10 ;    92   18   p =    2   ; R(n p^2 ) - p R(n) =  -2

      297 :     1    6   13    3    1    0
 count is 1 ; n =    1    2 ;     4    2   p =    2   ; R(n p^2 ) - p R(n) =  -2

      360 :     1    3   31    3    1    0
 count is 14 ; n =   37   20 ;  1813  124   p =    7   ; R(n p^2 ) - p R(n) =  -16

      392 :     3    3   12   -2    2    1
 count is 3 ; n =   13   10 ;   117   26   p =    3   ; R(n p^2 ) - p R(n) =  -4

      400 :     3    3   12    2    2    1
 count is 6 ; n =   23   10 ;   207   26   p =    3   ; R(n p^2 ) - p R(n) =  -4

      405 :     2    5   11    2    2    1
 count is 1 ; n =    2    2 ;    98    8   p =    7   ; R(n p^2 ) - p R(n) =  -6

      432 :     1    3   36    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

      432 :     1    3   37    3    1    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

      432 :     1    4   28    4    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

      432 :     1   12   12   12    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

      432 :     3    5    9    3    0    3
 count is 5 ; n =   17   10 ;   425   42   p =    5   ; R(n p^2 ) - p R(n) =  -8

      441 :     3    6    7    0    0    3
 count is 7 ; n =   31   20 ;   124   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      484 :     1    3   44    0    0    1
 count is 14 ; n =   53   16 ;   477   40   p =    3   ; R(n p^2 ) - p R(n) =  -8

      576 :     1    4   36    0    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

      600 :     5    7    7    6    5    5
 count is 10 ; n =   37   20 ;  1813  124   p =    7   ; R(n p^2 ) - p R(n) =  -16

      648 :     1    7   25    5    1    1
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

      675 :     1    4   45    0    0    1
 count is 12 ; n =   69   20 ;   276   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      675 :     5    6    6    3    0    0
 count is 5 ; n =   29   20 ;   116   36   p =    2   ; R(n p^2 ) - p R(n) =  -4

      720 :     3    5   15    3    3    3
 count is 1 ; n =    3    2 ;   147   10   p =    7   ; R(n p^2 ) - p R(n) =  -4

      768 :     1    8   24    0    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

      896 :     3    6   14    4    2    2
 count is 5 ; n =   19    6 ;   171   14   p =    3   ; R(n p^2 ) - p R(n) =  -4

      972 :     1    7   36    0    0    1
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     1024 :     1    8   32    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     1024 :     1   16   16    0    0    0
 count is 1 ; n =    1    2 ;     9    2   p =    3   ; R(n p^2 ) - p R(n) =  -4

     1024 :     3    3   32    0    0    2
 count is 12 ; n =   59   20 ;   531   52   p =    3   ; R(n p^2 ) - p R(n) =  -8

     1080 :     3    9   11    3    3    0
 count is 5 ; n =   17    8 ;   833   40   p =    7   ; R(n p^2 ) - p R(n) =  -16

     1125 :     1   10   29    5    1    0
 count is 1 ; n =    1    2 ;     4    2   p =    2   ; R(n p^2 ) - p R(n) =  -2

     1125 :     2    7   22   -6    1    1
 count is 1 ; n =    2    2 ;    98   12   p =    7   ; R(n p^2 ) - p R(n) =  -2

     1296 :     3    4   28    4    0    0
 count is 1 ; n =    3    2 ;    75    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     1323 :     2    8   21    0    0    1
 count is 5 ; n =   29   10 ;   116   18   p =    2   ; R(n p^2 ) - p R(n) =  -2

     1600 :     3    3   51   -2    2    2
 count is 11 ; n =   59   16 ;   531   40   p =    3   ; R(n p^2 ) - p R(n) =  -8

     1620 :     5    8   11   -4    1    2
 count is 2 ; n =    8    2 ;   392    8   p =    7   ; R(n p^2 ) - p R(n) =  -6

     1728 :     1   12   36    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     1792 :     5    8   12    0    4    0
 count is 12 ; n =   53   16 ;   477   32   p =    3   ; R(n p^2 ) - p R(n) =  -16

     1800 :     5   11   11    7    5    5
 count is 8 ; n =   41   16 ;  2009   80   p =    7   ; R(n p^2 ) - p R(n) =  -32

     2048 :     1    8   64    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     2160 :     5    9   15    9    3    3
 count is 2 ; n =    9    2 ;   441   10   p =    7   ; R(n p^2 ) - p R(n) =  -4

     2304 :     3    8   24    0    0    0
 count is 1 ; n =    3    2 ;    75    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

     2592 :     5    9   17    6    5    3
 count is 3 ; n =   17    6 ;   425   26   p =    5   ; R(n p^2 ) - p R(n) =  -4

     3072 :     1   16   48    0    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

     3375 :     2   15   32   15    1    0
 count is 3 ; n =   23    4 ;    92    4   p =    2   ; R(n p^2 ) - p R(n) =  -4

     3840 :     5    8   24    0    0    0
 count is 1 ; n =    5    2 ;   245    6   p =    7   ; R(n p^2 ) - p R(n) =  -8

     4032 :     7    8   20    0    4    4
 count is 3 ; n =   19    2 ;  2299   18   p =    11   ; R(n p^2 ) - p R(n) =  -4

     4096 :     3   11   32    0    0    2
 count is 8 ; n =   59   10 ;   531   26   p =    3   ; R(n p^2 ) - p R(n) =  -4

     4500 :     7    8   23    6    7    2
 count is 1 ; n =    8    2 ;   392   12   p =    7   ; R(n p^2 ) - p R(n) =  -2

     4536 :     5    9   27    0    3    3
 count is 8 ; n =   41    6 ;  1025   22   p =    5   ; R(n p^2 ) - p R(n) =  -8

     5120 :     7   12   16    0    0    4
 count is 12 ; n =   79   16 ;   711   32   p =    3   ; R(n p^2 ) - p R(n) =  -16

     5184 :     5    8   36    0    0    4
 count is 7 ; n =   41    6 ;  1025   26   p =    5   ; R(n p^2 ) - p R(n) =  -4

     5400 :     7    7   28   -2    2    1
 count is 4 ; n =   37   10 ;  1813   62   p =    7   ; R(n p^2 ) - p R(n) =  -8

     6912 :     1   16  112   16    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

     6912 :     1   24   72    0    0    0
 count is 1 ; n =    1    2 ;    25    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

     6912 :     1   48   48   48    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     6912 :     4   13   37    2    4    4
 count is 1 ; n =    4    2 ;   100    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

     6912 :     5    5   72    0    0    2
 count is 11 ; n =   77   16 ;  1925   64   p =    5   ; R(n p^2 ) - p R(n) =  -16

     6912 :     8    9   24    0    0    0
 count is 2 ; n =    9    2 ;   225    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

     8232 :     5   13   33   -6    3    1
 count is 14 ; n =   73    6 ;  1825   24   p =    5   ; R(n p^2 ) - p R(n) =  -6

     8448 :     7   15   23   -6    2    6
 count is 8 ; n =   47    6 ;  1175   26   p =    5   ; R(n p^2 ) - p R(n) =  -4

     9216 :     3   16   48    0    0    0
 count is 1 ; n =    3    2 ;    75    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

    10125 :     9   11   29   -4    3    6
 count is 2 ; n =   14    2 ;  1694   18   p =    11   ; R(n p^2 ) - p R(n) =  -4

    11520 :     8   15   24    0    0    0
 count is 2 ; n =   15    2 ;   735    6   p =    7   ; R(n p^2 ) - p R(n) =  -8

    11520 :    11   16   19    8    2    8
 count is 2 ; n =   19    4 ;  2299   36   p =    11   ; R(n p^2 ) - p R(n) =  -8

    12544 :     3   19   56    0    0    2
 count is 11 ; n =   83   10 ;   747   26   p =    3   ; R(n p^2 ) - p R(n) =  -4

    19200 :     1   40  120    0    0    0
 count is 1 ; n =    1    2 ;    49    6   p =    7   ; R(n p^2 ) - p R(n) =  -8

    20736 :     3   16  112   16    0    0
 count is 1 ; n =    3    2 ;    75    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

    20736 :     7   15   55   -6    2    6
 count is 8 ; n =   79    8 ;  1975   36   p =    5   ; R(n p^2 ) - p R(n) =  -4

    24696 :    11   15   39   -3    6    3
 count is 7 ; n =   53    4 ;  1325   16   p =    5   ; R(n p^2 ) - p R(n) =  -4

    25344 :     5   20   68   -8    4    4
 count is 12 ; n =  101   10 ;  2525   34   p =    5   ; R(n p^2 ) - p R(n) =  -16

    25600 :     3   27   80    0    0    2
 count is 9 ; n =  107   10 ;   963   26   p =    3   ; R(n p^2 ) - p R(n) =  -4

    27648 :     1   48  144    0    0    0
 count is 1 ; n =    1    2 ;    25    2   p =    5   ; R(n p^2 ) - p R(n) =  -8

    27648 :     5   20   77   20    2    4
 count is 6 ; n =   77    8 ;  1925   32   p =    5   ; R(n p^2 ) - p R(n) =  -8

    27648 :     9   17   48    0    0    6
 count is 1 ; n =    9    2 ;   225    6   p =    5   ; R(n p^2 ) - p R(n) =  -4

    32000 :    11   16   51    8    2    8
 count is 9 ; n =   91    8 ;   819   20   p =    3   ; R(n p^2 ) - p R(n) =  -4

    34560 :    13   24   28    0    4    0
 count is 3 ; n =   37    6 ;  1813   30   p =    7   ; R(n p^2 ) - p R(n) =  -12

    57600 :     3   40  120    0    0    0
 count is 1 ; n =    3    2 ;   147    6   p =    7   ; R(n p^2 ) - p R(n) =  -8

    57600 :     7   23   92   12    4    2
 count is 13 ; n =  143    8 ;  7007   48   p =    7   ; R(n p^2 ) - p R(n) =  -8

   172800 :     9   41  120    0    0    6
 count is 1 ; n =    9    2 ;   441    6   p =    7   ; R(n p^2 ) - p R(n) =  -8

jagy@phobeusjunior:  grep " p = "  Class_Number_NOT_One.txt | wc   
    119    2737    9684
jagy@phobeusjunior:    date
Sat Sep  6 10:51:20 PDT 2014

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Later Saturday: I really, really like this one. In case anyone bites, related questions include: for ternary forms, to what extent does class number one (the 794) give these nice formulas as in Hecke eigenforms? I am capable of gathering evidence for and against that myself, I am guessing the only wiggle room is primes dividing the discriminant.

In higher dimension: what can one say about higher theta series? For (positive) quaternary forms, we would be talking about counting representations of numbers, but then we would count representations of binary forms, for convenience i would have them Gauss reduced which is easy. Andrew Earnest initiated the study of quaternaries representing binaries and showed there are finitely many "2-regular" quaternaries. Meanwhile, Watson showed that class number one occurs only up to dimension 10. Quite recently, Nebe's students Lorch and Kirschmer compiled a reliable list of all class number one forms. Can we make a theory of theta series for, especially, codimension 2 representations, that includes the Hecke operator stuff, and relate that to class number one?

Certainly seems to be a nice theory of such higher theta series, the key phrase seems to be "Siegel modular forms," see http://arxiv.org/abs/1202.4909 and other stuff by Schulze-Pillot, also for variety http://www.math.snu.ac.kr/~mhkim/thesis/thesis_33.pdf

Sunday; upon further reflection, I suspect that Siegel's weighted average representation over a genus always obeys these sorts of laws, not something I knew. With class number one, the theta series for the form agrees with that of the genus, hence explaining one direction of this pretty well. It also justifies the prevalent practice, for even dimension 4 or larger, of identifying forms with improper ( determinant $-1$) equivalence, because they represent the same numbers (or lower dimension quadratic forms) with the same representation counts.

Also, there is no real problem replacing a number $n$ with an equivalence class $N$ of lower dimensional quadratic forms/lattices; this is how Schulze-Pillot writes it. I have enough software written to experiment with quaternaries representing binaries. I am not sure about other codimension; quaternaries representing numbers is codimension 3, maybe the standard relation (from Jeremy's answer) changes form in codimension other than 2.

Found a nice summary pdf by Winfried Kohnen. Don't see any explicit laws on representation counts, maybe reading between the lines. There are quite a number of relevant items by Lynne Walling

Answer 1

There are many specific questions raised in this post. I will address the main one and show that if a $Q$ is a ternary quadratic form and the theta series $\theta_{Q}$ is a Hecke eigenform, then $Q$ has class number one.

The form $\theta_{Q}$ is a weight $3/2$ modular form and any such form has a decomposition as $E + H + C$. Here $E$ is an Eisenstein series whose coefficients are specified by the local densities of $Q$. The form $H$ is a linear combination of "unary theta series": forms of the shape $\sum \psi(n) n q^{tn^{2}}$, where $\psi(n)$ is an odd Dirichlet character. The form $C$ is a weight $3/2$ cusp form whose Shimura lifts are also a cusp forms. Each piece is orthogonal to the other two under the Petersson inner product. Because the Hecke operators are self-adjoint (or close to it) with respect to this inner product, we have that $\theta_{Q} | T_{p^{2}} = \lambda \theta_{Q}$ implies $E | T_{p^{2}} = \lambda E$, $H | T_{p^{2}} = \lambda H$, and $C | T_{p^{2}} = \lambda C$.

By looking at the formulas for local densities, I've convinced myself that if $p$ does not divide the discriminant of $Q$, then $E$ is an eigenfunction for the half-integer weight Hecke operator $T_{p^{2}}$ with eigenvalue $p + 1 - \left(\frac{-\Delta}{p}\right)$.

On the other hand, $C$ has a decomposition into Hecke eigenforms each of whose Shimura lifts are weight $2$ newforms. These Hecke eigenvalues are bounded in absolute value by $2 \sqrt{p}$.

Finally, if $g(z) = \sum_{n=1}^{\infty} \psi(n) n q^{tn^{2}}$, then $g(z) | T_{p^{2}} = \psi(p) (p+1) g(z)$.

It follows that since each of the pieces $E$, $H$ and $C$ must be a Hecke eigenform with the same eigenvalue $\lambda_{p}$, the Hecke eigenvalues are incompatible unless $H = C = 0$.

The upshot is that if $\theta_{Q}$ is a Hecke eigenform then $H = C = 0$. Hence $\theta_{Q} = E$, which implies immediately that $Q$ is regular. Based on Will's computations for the regular ternary quadratic forms $Q$ the only $Q$ with $\theta_{Q}$ a Hecke eigenform are those $Q$ with class number one.

I don't see an a priori reason why $H = C = 0$ implies class number one. It seems possible to me that there might be forms $Q$ in more variables that are regular, with class number greater than one, for which $C = 0$. (The $H$ term in the decomposition only appears for ternary forms.) I'm not sure what to expect for higher theta series - I would guess that the Hecke action is more complicated.