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There are some half-integral weight modular forms like Jacobi's theta functions and it can be interpreted as automorphic representations on metaplectic groups (double cover of symplectic groups). I always wondered why there's no 1/3-weight modular forms, and I found that many people already asked about it in MO. So what I found is that Patterson constructed an analogue of theta function for 3-fold cover of $\mathrm{GL}_{2}$, which is used to prove equidistributionality of angular components of cubic Gauss sums (Heath-Brown-Patterson).

What suddenly come to my mind is to construct such modular forms of rational weight as a formal series, and proving its modularity later. For example, the most celebrated example, $$ \theta(z) = \sum_{n\in \mathbb{Z}} q^{n^2} = 1 + 2q + 2q^{4} + 2q^{9} + \cdots $$ is a 1/2-weight modular form of level 4, and we can formally take square root of this to get $$ \theta(z)^{1/2} = (1 + 2q + 2q^{4} + 2q^{9} + \cdots)^{1/2} = 1 + \sum_{n\geq 1}b_{n}q^{n} $$ for some $\{b_{n}\}_{n \geq 1}$, and expect this to be a weight 1/4 modular form. We can compute $b_{n}$ and the first few terms are (if I wrote the code correctly) $$ 1 + q - \frac{1}{2}q^{2}+ \frac{1}{2}q^{3} + \frac{3}{8}q^{4} - \frac{1}{8}q^{5}+ \frac{3}{16}q^{6} - \frac{7}{16}q^{7} + \frac{67}{128}q^{8}+ \frac{27}{128}q^{9} + \cdots $$ The denominators are powers of 2 but numerators seem to me quite random (I couldn't find it on OEIS). I don't know how to prove the modularity of the series (I don't even know whether it is or not, with respect to some discrete subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$), so I followed this answer to plot the graph and see if I can find any symmetry, and here's a graph: (use degree 1000 approximation)

enter image description here

It is hard for me to find any symmetry through this image. I wonder if there's a possibility that the series is modular with respect to some discrete $\Gamma \leq \mathrm{SL}_{2}(\mathbb{Z})$, and wonder if we can actually prove it. Also, if it is true, I wonder if it can be extended to other weights, as $\theta(z)^{2p/q}$ for weight $p/q$-modular forms.

FYI: here's a plot for the theta function:

enter image description here

FYI: here's a list of coefficients $b_{n}$ for $n \leq 100$:

0: 1
1: 1
2: -1/2
3: 1/2
4: 3/8
5: -1/8
6: 3/16
7: -7/16
8: 67/128
9: 27/128
10: 49/256
11: -41/256
12: -121/1024
13: 11/1024
14: -469/2048
15: 1161/2048
16: 8419/32768
17: 4115/32768
18: -23835/65536
19: 3051/65536
20: -50259/262144
21: 59073/262144
22: 70693/524288
23: -89009/524288
24: 460839/4194304
25: 829071/4194304
26: 1295653/8388608
27: -144813/8388608
28: 6202511/33554432
29: -8155293/33554432
30: -276229/67108864
31: -19726295/67108864
32: 380954275/2147483648
33: 7422723/2147483648
34: 82356845/4294967296
35: 1111214643/4294967296
36: 5918448465/17179869184
37: -2478914555/17179869184
38: 1292307073/34359738368
39: 963306739/34359738368
40: -40242306771/274877906944
41: 939841653/274877906944
42: -70323248625/549755813888
43: 31230714057/549755813888
44: -350589344511/2199023255552
45: 429098830461/2199023255552
46: 158556614893/4398046511104
47: -162965502305/4398046511104
48: 12513716236295/70368744177664
49: 33212258501207/70368744177664
50: -28075611555343/140737488355328
51: -12588929453889/140737488355328
52: -86216253046575/562949953421312
53: 24096598963477/562949953421312
54: -5426269169703/1125899906842624
55: 236228228614667/1125899906842624
56: 208302650849391/9007199254740992
57: -2153357177794697/9007199254740992
58: -146753418010803/18014398509481984
59: -214469519444341/18014398509481984
60: 1690652329895583/72057594037927936
61: 2011055311332115/72057594037927936
62: -10484156572473189/144115188075855872
63: 16598673444331881/144115188075855872
64: 2457431928589435939/9223372036854775808
65: 220277769549083875/9223372036854775808
66: 5393177533349245949/18446744073709551616
67: -1190858968231218877/18446744073709551616
68: 3692310809391496889/73786976294838206464
69: -15780224336921375571/73786976294838206464
70: -26992678599519487463/147573952589676412928
71: -20686281813588373157/147573952589676412928
72: 200604885788920693137/1180591620717411303424
73: -20177990263729151591/1180591620717411303424
74: -78352192599846905077/2361183241434822606848
75: -123727125386079951475/2361183241434822606848
76: -206603346880948591955/9444732965739290427392
77: 916574102650931294745/9444732965739290427392
78: 5325754040578459796505/18889465931478580854784
79: -1884087088589718283549/18889465931478580854784
80: 38530470313914699307789/302231454903657293676544
81: 33836564472999596867101/302231454903657293676544
82: -32705764563608726531189/604462909807314587353088
83: 11991501594688003256677/604462909807314587353088
84: -111531452293531549054573/2417851639229258349412352
85: 345651893208147993211455/2417851639229258349412352
86: 33481759452214692139579/4835703278458516698824704
87: -345134572175266995740271/4835703278458516698824704
88: -7280666442338344325124895/38685626227668133590597632
89: 1332492381396426007334265/38685626227668133590597632
90: -6137972576875331481316653/77371252455336267181195264
91: 9278551312517725479692213/77371252455336267181195264
92: 19281264251724365555614121/309485009821345068724781056
93: -41048881513866151425532235/309485009821345068724781056
94: -43434756598231640063507459/618970019642690137449562112
95: 81746311552513874789685439/618970019642690137449562112
96: 464636279630891564852366919/19807040628566084398385987584
97: 3338041883135802049476823271/19807040628566084398385987584
98: -10614400742548410689935125207/39614081257132168796771975168
99: 9701698932455472357764821815/39614081257132168796771975168
100: 31109958365571876785493506093/158456325028528675187087900672
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    $\begingroup$ "I expect this to be a weight 1/4 modular form" – this statement cannot be proved or disproved until you make precise what the words mean. I think that's kind of the problem here. What drives the theory of weight 1/2 modular forms, and the various generalisations studied by Patterson, Brylinski–Deligne etc, is the availability of a natural choice of square root of the automorphy factor $j(\gamma, z)$. $\endgroup$
    – David Loeffler
    Commented Sep 3, 2022 at 10:44
  • $\begingroup$ As far as I'm aware there isn't a natural choice of fourth root. You could simply choose the fourth root that makes the transformation law work for your function, so you would be setting up the definitions so your function is modular by definition; but then the burden is on you to show that this is a mathematically interesting concept. $\endgroup$
    – David Loeffler
    Commented Sep 3, 2022 at 10:46
  • $\begingroup$ It's an interesting idea, but do you have a reason to believe the Jacobi theta series should have a nice modular square root? E.g., if you just take square roots of random theta series/modular forms of weight $2k$, you won't typically get modular forms of weight $k$. (This follows from dimension formulas.) $\endgroup$
    – Kimball
    Commented Sep 3, 2022 at 13:47
  • $\begingroup$ @DavidLoeffler I agree with you. Maybe I'll do some more experiments to make the conjecture more precise with various choices of $j(\gamma, z)^{1/2}$. $\endgroup$
    – Seewoo Lee
    Commented Sep 3, 2022 at 14:51
  • $\begingroup$ @Kimball With your comment, I just realized that even simpler example for integral weights is not working - there's no weight 6 modular form whose square is $\Delta(z)$ (maybe there is with higher weights - I never heard of it actually). In honest, there's no strong reason to believe this to be true, but this was somewhat natural candidate that I can apply my naive idea. $\endgroup$
    – Seewoo Lee
    Commented Sep 3, 2022 at 14:54

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