You have a good answer; here are some computations about **primitive** representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$   I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.

Made a little summary: the ratios $(r_0/h(4M), r_1/h(4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$




    ---------------------------------------
     g0  :          9 :     1    1    3    0    0    1 auto 24
     g1  :         36 :     1    3    3    0    0    0 auto 16
    ---------------------------------------
    ---------------------------------------
    
          M     r0   r1  h(4M)  %72  r0/h r1/h
          1      6    2      1  N 1    6    2           1 =  1 
          3      8    4      1  N 3    8    4           3 = 3
          4     12    8      1  N 4   12    8           4 = 2^2
          6     12    4      2  N 6    6    2           6 = 2 * 3
          7     24   16      1  N 7   24   16           7 = 7
         10     24    8      2  N10   12    4          10 = 2 * 5
         12     12    8      2  N12    6    4          12 = 2^2 * 3
         13     24    8      2  N13   12    4          13 = 13
         15     24   16      2  N15   12    8          15 = 3 * 5
         16     24   16      2  N16   12    8          16 = 2^4
         19     48   24      3  N19   16    8          19 = 19
         21     24    8      4  N21    6    2          21 = 3 * 7
         22     24    8      2  N22   12    4          22 = 2 * 11
    -------------------------------------------------------------------------
          1      6    2      1  N 1    6    2           1 =  1 
         73     48   16      4  N 1   12    4          73 = 73
        145     96   32      8  N 1   12    4         145 = 5 * 29
        217     96   32      8  N 1   12    4         217 = 7 * 31
        289     96   32      8  N 1   12    4         289 = 17^2
        361    120   40     10  N 1   12    4         361 = 19^2
        433    144   48     12  N 1   12    4         433 = 433
        505     96   32      8  N 1   12    4         505 = 5 * 101
        577     96   32      8  N 1   12    4         577 = 577
        649    240   80     20  N 1   12    4         649 = 11 * 59
        721    192   64     16  N 1   12    4         721 = 7 * 103
        793     96   32      8  N 1   12    4         793 = 13 * 61
        865    192   64     16  N 1   12    4         865 = 5 * 173
        937    240   80     20  N 1   12    4         937 = 937
       1009    240   80     20  N 1   12    4        1009 = 1009
       1081    192   64     16  N 1   12    4        1081 = 23 * 47
       1153    192   64     16  N 1   12    4        1153 = 1153
       1225    192   64     16  N 1   12    4        1225 = 5^2 * 7^2
       1297    144   48     12  N 1   12    4        1297 = 1297
       1369    216   72     18  N 1   12    4        1369 = 37^2
    ------------------------------------------------------------------------
          7     24   16      1  N 7   24   16           7 = 7
         79    120   80      5  N 7   24   16          79 = 79
        151    168  112      7  N 7   24   16         151 = 151
        223    168  112      7  N 7   24   16         223 = 223
        295    192  128      8  N 7   24   16         295 = 5 * 59
        367    216  144      9  N 7   24   16         367 = 367
        439    360  240     15  N 7   24   16         439 = 439
        511    336  224     14  N 7   24   16         511 = 7 * 73
        583    192  128      8  N 7   24   16         583 = 11 * 53
        655    288  192     12  N 7   24   16         655 = 5 * 131
        727    312  208     13  N 7   24   16         727 = 727
        799    384  256     16  N 7   24   16         799 = 17 * 47
        871    528  352     22  N 7   24   16         871 = 13 * 67
        943    384  256     16  N 7   24   16         943 = 23 * 41
       1015    384  256     16  N 7   24   16        1015 = 5 * 7 * 29
       1087    216  144      9  N 7   24   16        1087 = 1087
       1159    384  256     16  N 7   24   16        1159 = 19 * 61
       1231    648  432     27  N 7   24   16        1231 = 1231
       1303    264  176     11  N 7   24   16        1303 = 1303
       1375    480  320     20  N 7   24   16        1375 = 5^3 * 11
    ------------------------------------------------------------------
         13     24    8      2  N13   12    4          13 = 13
         85     48   16      4  N13   12    4          85 = 5 * 17
        157     72   24      6  N13   12    4         157 = 157
        229    120   40     10  N13   12    4         229 = 229
        301     96   32      8  N13   12    4         301 = 7 * 43
        373    120   40     10  N13   12    4         373 = 373
        445     96   32      8  N13   12    4         445 = 5 * 89
        517    144   48     12  N13   12    4         517 = 11 * 47
        589    192   64     16  N13   12    4         589 = 19 * 31
        661    216   72     18  N13   12    4         661 = 661
        733    168   56     14  N13   12    4         733 = 733
        805    192   64     16  N13   12    4         805 = 5 * 7 * 23
        877    120   40     10  N13   12    4         877 = 877
        949    144   48     12  N13   12    4         949 = 13 * 73
       1021    264   88     22  N13   12    4        1021 = 1021
       1093    120   40     10  N13   12    4        1093 = 1093
       1165    240   80     20  N13   12    4        1165 = 5 * 233
       1237    168   56     14  N13   12    4        1237 = 1237
       1309    288   96     24  N13   12    4        1309 = 7 * 11 * 17
       1381    312  104     26  N13   12    4        1381 = 1381
    ------------------------------------------------------------------
         19     48   24      3  N19   16    8          19 = 19
         91     96   48      6  N19   16    8          91 = 7 * 13
        163     48   24      3  N19   16    8         163 = 163
        235     96   48      6  N19   16    8         235 = 5 * 47
        307    144   72      9  N19   16    8         307 = 307
        379    144   72      9  N19   16    8         379 = 379
        451    288  144     18  N19   16    8         451 = 11 * 41
        523    240  120     15  N19   16    8         523 = 523
        595    192   96     12  N19   16    8         595 = 5 * 7 * 17
        667    192   96     12  N19   16    8         667 = 23 * 29
        739    240  120     15  N19   16    8         739 = 739
        811    336  168     21  N19   16    8         811 = 811
        883    144   72      9  N19   16    8         883 = 883
        955    192   96     12  N19   16    8         955 = 5 * 191
       1027    192   96     12  N19   16    8        1027 = 13 * 79
       1099    288  144     18  N19   16    8        1099 = 7 * 157
       1171    336  168     21  N19   16    8        1171 = 1171
       1243    192   96     12  N19   16    8        1243 = 11 * 113
       1315    288  144     18  N19   16    8        1315 = 5 * 263
       1387    192   96     12  N19   16    8        1387 = 19 * 73
    ------------------------------------------------------------------
         25     24    8      2  N25   12    4          25 = 5^2
         97     48   16      4  N25   12    4          97 = 97
        169     72   24      6  N25   12    4         169 = 13^2
        241    144   48     12  N25   12    4         241 = 241
        313     96   32      8  N25   12    4         313 = 313
        385     96   32      8  N25   12    4         385 = 5 * 7 * 11
        457     96   32      8  N25   12    4         457 = 457
        529    144   48     12  N25   12    4         529 = 23^2
        601    240   80     20  N25   12    4         601 = 601
        673    144   48     12  N25   12    4         673 = 673
        745    192   64     16  N25   12    4         745 = 5 * 149
        817    144   48     12  N25   12    4         817 = 19 * 43
        889    192   64     16  N25   12    4         889 = 7 * 127
        961    192   64     16  N25   12    4         961 = 31^2
       1033    144   48     12  N25   12    4        1033 = 1033
       1105    192   64     16  N25   12    4        1105 = 5 * 13 * 17
       1177    144   48     12  N25   12    4        1177 = 11 * 107
       1249    384  128     32  N25   12    4        1249 = 1249
       1321    288   96     24  N25   12    4        1321 = 1321
       1393    192   64     16  N25   12    4        1393 = 7 * 199
    -------------------------------------------------------------------
         31     72   48      3  N31   24   16          31 = 31
        103    120   80      5  N31   24   16         103 = 103
        175    144   96      6  N31   24   16         175 = 5^2 * 7
        247    144   96      6  N31   24   16         247 = 13 * 19
        319    240  160     10  N31   24   16         319 = 11 * 29
        391    336  224     14  N31   24   16         391 = 17 * 23
        463    168  112      7  N31   24   16         463 = 463
        535    336  224     14  N31   24   16         535 = 5 * 107
        607    312  208     13  N31   24   16         607 = 607
        679    432  288     18  N31   24   16         679 = 7 * 97
        751    360  240     15  N31   24   16         751 = 751
        823    216  144      9  N31   24   16         823 = 823
        895    384  256     16  N31   24   16         895 = 5 * 179
        967    264  176     11  N31   24   16         967 = 967
       1039    552  368     23  N31   24   16        1039 = 1039
       1111    528  352     22  N31   24   16        1111 = 11 * 101
       1183    336  224     14  N31   24   16        1183 = 7 * 13^2
       1255    288  192     12  N31   24   16        1255 = 5 * 251
       1327    360  240     15  N31   24   16        1327 = 1327
       1399    648  432     27  N31   24   16        1399 = 1399
    ---------------------------------------------------------------
         37     24    8      2  N37   12    4          37 = 37
        109     72   24      6  N37   12    4         109 = 109
        181    120   40     10  N37   12    4         181 = 181
        253     48   16      4  N37   12    4         253 = 11 * 23
        325    144   48     12  N37   12    4         325 = 5^2 * 13
        397     72   24      6  N37   12    4         397 = 397
        469    192   64     16  N37   12    4         469 = 7 * 67
        541    120   40     10  N37   12    4         541 = 541
        613    120   40     10  N37   12    4         613 = 613
        685    144   48     12  N37   12    4         685 = 5 * 137
        757    120   40     10  N37   12    4         757 = 757
        829    264   88     22  N37   12    4         829 = 829
        901    288   96     24  N37   12    4         901 = 17 * 53
        973    144   48     12  N37   12    4         973 = 7 * 139
       1045    192   64     16  N37   12    4        1045 = 5 * 11 * 19
       1117    168   56     14  N37   12    4        1117 = 1117
       1189    240   80     20  N37   12    4        1189 = 29 * 41
       1261    240   80     20  N37   12    4        1261 = 13 * 97
       1333    240   80     20  N37   12    4        1333 = 31 * 43
       1405    288   96     24  N37   12    4        1405 = 5 * 281
    ------------------------------------------------------------------
         43     48   24      3  N43   16    8          43 = 43
        115     96   48      6  N43   16    8         115 = 5 * 23
        187     96   48      6  N43   16    8         187 = 11 * 17
        259    192   96     12  N43   16    8         259 = 7 * 37
        331    144   72      9  N43   16    8         331 = 331
        403     96   48      6  N43   16    8         403 = 13 * 31
        475    192   96     12  N43   16    8         475 = 5^2 * 19
        547    144   72      9  N43   16    8         547 = 547
        619    240  120     15  N43   16    8         619 = 619
        691    240  120     15  N43   16    8         691 = 691
        763    192   96     12  N43   16    8         763 = 7 * 109
        835    288  144     18  N43   16    8         835 = 5 * 167
        907    144   72      9  N43   16    8         907 = 907
        979    384  192     24  N43   16    8         979 = 11 * 89
       1051    240  120     15  N43   16    8        1051 = 1051
       1123    240  120     15  N43   16    8        1123 = 1123
       1195    384  192     24  N43   16    8        1195 = 5 * 239
       1267    288  144     18  N43   16    8        1267 = 7 * 181
       1339    384  192     24  N43   16    8        1339 = 13 * 103
       1411    192   96     12  N43   16    8        1411 = 17 * 83
    -------------------------------------------------------------------
         49     48   16      4  N49   12    4          49 = 7^2
        121     72   24      6  N49   12    4         121 = 11^2
        193     48   16      4  N49   12    4         193 = 193
        265     96   32      8  N49   12    4         265 = 5 * 53
        337     96   32      8  N49   12    4         337 = 337
        409    192   64     16  N49   12    4         409 = 409
        481    192   64     16  N49   12    4         481 = 13 * 37
        553     96   32      8  N49   12    4         553 = 7 * 79
        625    120   40     10  N49   12    4         625 = 5^4
        697     96   32      8  N49   12    4         697 = 17 * 41
        769    240   80     20  N49   12    4         769 = 769
        841    168   56     14  N49   12    4         841 = 29^2
        913    144   48     12  N49   12    4         913 = 11 * 83
        985    288   96     24  N49   12    4         985 = 5 * 197
       1057    192   64     16  N49   12    4        1057 = 7 * 151
       1129    192   64     16  N49   12    4        1129 = 1129
       1201    192   64     16  N49   12    4        1201 = 1201
       1273    240   80     20  N49   12    4        1273 = 19 * 67
       1345    192   64     16  N49   12    4        1345 = 5 * 269
       1417    192   64     16  N49   12    4        1417 = 13 * 109
    ------------------------------------------------------------------
         55     96   64      4  N55   24   16          55 = 5 * 11
        127    120   80      5  N55   24   16         127 = 127
        199    216  144      9  N55   24   16         199 = 199
        271    264  176     11  N55   24   16         271 = 271
        343    168  112      7  N55   24   16         343 = 7^3
        415    240  160     10  N55   24   16         415 = 5 * 83
        487    168  112      7  N55   24   16         487 = 487
        559    384  256     16  N55   24   16         559 = 13 * 43
        631    312  208     13  N55   24   16         631 = 631
        703    336  224     14  N55   24   16         703 = 19 * 37
        775    288  192     12  N55   24   16         775 = 5^2 * 31
        847    240  160     10  N55   24   16         847 = 7 * 11^2
        919    456  304     19  N55   24   16         919 = 919
        991    408  272     17  N55   24   16         991 = 991
       1063    456  304     19  N55   24   16        1063 = 1063
       1135    432  288     18  N55   24   16        1135 = 5 * 227
       1207    432  288     18  N55   24   16        1207 = 17 * 71
       1279    552  368     23  N55   24   16        1279 = 1279
       1351    576  384     24  N55   24   16        1351 = 7 * 193
       1423    216  144      9  N55   24   16        1423 = 1423
    --------------------------------------------------------------
         61     72   24      6  N61   12    4          61 = 61
        133     48   16      4  N61   12    4         133 = 7 * 19
        205     96   32      8  N61   12    4         205 = 5 * 41
        277     72   24      6  N61   12    4         277 = 277
        349    168   56     14  N61   12    4         349 = 349
        421    120   40     10  N61   12    4         421 = 421
        493    144   48     12  N61   12    4         493 = 17 * 29
        565    144   48     12  N61   12    4         565 = 5 * 113
        637    144   48     12  N61   12    4         637 = 7^2 * 13
        709    120   40     10  N61   12    4         709 = 709
        781    240   80     20  N61   12    4         781 = 11 * 71
        853    120   40     10  N61   12    4         853 = 853
        925    144   48     12  N61   12    4         925 = 5^2 * 37
        997    168   56     14  N61   12    4         997 = 997
       1069    360  120     30  N61   12    4        1069 = 1069
       1141    288   96     24  N61   12    4        1141 = 7 * 163
       1213    120   40     10  N61   12    4        1213 = 1213
       1285    144   48     12  N61   12    4        1285 = 5 * 257
       1357    192   64     16  N61   12    4        1357 = 23 * 59
       1429    264   88     22  N61   12    4        1429 = 1429
    ----------------------------------------------------------------
         67     48   24      3  N67   16    8          67 = 67
        139    144   72      9  N67   16    8         139 = 139
        211    144   72      9  N67   16    8         211 = 211
        283    144   72      9  N67   16    8         283 = 283
        355    192   96     12  N67   16    8         355 = 5 * 71
        427     96   48      6  N67   16    8         427 = 7 * 61
        499    144   72      9  N67   16    8         499 = 499
        571    240  120     15  N67   16    8         571 = 571
        643    144   72      9  N67   16    8         643 = 643
        715    192   96     12  N67   16    8         715 = 5 * 11 * 13
        787    240  120     15  N67   16    8         787 = 787
        859    336  168     21  N67   16    8         859 = 859
        931    288  144     18  N67   16    8         931 = 7^2 * 19
       1003    192   96     12  N67   16    8        1003 = 17 * 59
       1075    288  144     18  N67   16    8        1075 = 5^2 * 43
       1147    288  144     18  N67   16    8        1147 = 31 * 37
       1219    288  144     18  N67   16    8        1219 = 23 * 53
       1291    432  216     27  N67   16    8        1291 = 1291
       1363    288  144     18  N67   16    8        1363 = 29 * 47
       1435    192   96     12  N67   16    8        1435 = 5 * 7 * 41
    -----------------------------------------------------------------------
          M     r0   r1  h(4M)  %72  r0/h r1/h
    ---------------------------------------
     g0  :          9 :     1    1    3    0    0    1 auto 24
     g1  :         36 :     1    3    3    0    0    0 auto 16
    ---------------------------------------
    ---------------------------------------

      M     r0   r1  h(4M)  %72  r0/h r1/h