# Properties

 Label 396.4 Level 396 Weight 4 Dimension 5805 Nonzero newspaces 16 Sturm bound 34560 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$396 = 2^{2} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$34560$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(396))$$.

Total New Old
Modular forms 13360 5969 7391
Cusp forms 12560 5805 6755
Eisenstein series 800 164 636

## Trace form

 $$5805 q - 9 q^{2} + 6 q^{3} - 37 q^{4} - 66 q^{5} - 62 q^{6} - 14 q^{7} - 15 q^{8} - 130 q^{9} + O(q^{10})$$ $$5805 q - 9 q^{2} + 6 q^{3} - 37 q^{4} - 66 q^{5} - 62 q^{6} - 14 q^{7} - 15 q^{8} - 130 q^{9} + 264 q^{10} + 35 q^{11} - 28 q^{12} - 154 q^{13} + 316 q^{14} + 360 q^{15} - 361 q^{16} + 280 q^{17} + 220 q^{18} - 85 q^{19} - 22 q^{20} + 452 q^{21} - 180 q^{22} + 404 q^{23} - 26 q^{24} + 358 q^{25} - 40 q^{26} - 1356 q^{27} - 216 q^{28} - 2116 q^{29} - 896 q^{30} - 274 q^{31} - 1404 q^{32} - 641 q^{33} - 948 q^{34} + 260 q^{35} - 2066 q^{36} - 1102 q^{37} - 1472 q^{38} + 372 q^{39} + 570 q^{40} - 642 q^{41} - 1564 q^{42} - 32 q^{43} + 1490 q^{44} - 1216 q^{45} + 234 q^{46} - 520 q^{47} + 2722 q^{48} + 4364 q^{49} + 2099 q^{50} + 1844 q^{51} - 2584 q^{52} + 2414 q^{53} + 662 q^{54} + 134 q^{55} - 28 q^{56} - 114 q^{57} + 402 q^{58} + 651 q^{59} + 7484 q^{60} - 4066 q^{61} + 11910 q^{62} + 928 q^{63} + 12299 q^{64} + 2820 q^{65} + 5316 q^{66} + 5652 q^{67} + 1800 q^{68} + 4176 q^{69} - 518 q^{70} + 4160 q^{71} - 7520 q^{72} + 3564 q^{73} - 15608 q^{74} - 3592 q^{75} - 6220 q^{76} - 8978 q^{77} - 14108 q^{78} - 8098 q^{79} - 15710 q^{80} - 19026 q^{81} - 10331 q^{82} - 10647 q^{83} + 8250 q^{84} - 3734 q^{85} + 20167 q^{86} - 628 q^{87} + 9294 q^{88} + 1178 q^{89} + 22384 q^{90} - 2772 q^{91} + 24078 q^{92} + 10312 q^{93} + 13654 q^{94} + 18978 q^{95} + 8050 q^{96} + 13977 q^{97} + 17534 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(396))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
396.4.a $$\chi_{396}(1, \cdot)$$ 396.4.a.a 1 1
396.4.a.b 1
396.4.a.c 1
396.4.a.d 1
396.4.a.e 1
396.4.a.f 1
396.4.a.g 1
396.4.a.h 2
396.4.a.i 2
396.4.a.j 2
396.4.b $$\chi_{396}(197, \cdot)$$ 396.4.b.a 6 1
396.4.b.b 6
396.4.c $$\chi_{396}(287, \cdot)$$ 396.4.c.a 30 1
396.4.c.b 30
396.4.h $$\chi_{396}(307, \cdot)$$ 396.4.h.a 4 1
396.4.h.b 16
396.4.h.c 32
396.4.h.d 36
396.4.i $$\chi_{396}(133, \cdot)$$ 396.4.i.a 6 2
396.4.i.b 22
396.4.i.c 32
396.4.j $$\chi_{396}(37, \cdot)$$ 396.4.j.a 4 4
396.4.j.b 8
396.4.j.c 12
396.4.j.d 12
396.4.j.e 24
396.4.k $$\chi_{396}(43, \cdot)$$ n/a 424 2
396.4.p $$\chi_{396}(23, \cdot)$$ n/a 360 2
396.4.q $$\chi_{396}(65, \cdot)$$ 396.4.q.a 4 2
396.4.q.b 4
396.4.q.c 64
396.4.r $$\chi_{396}(19, \cdot)$$ n/a 352 4
396.4.w $$\chi_{396}(71, \cdot)$$ n/a 288 4
396.4.x $$\chi_{396}(17, \cdot)$$ 396.4.x.a 48 4
396.4.y $$\chi_{396}(25, \cdot)$$ n/a 288 8
396.4.z $$\chi_{396}(29, \cdot)$$ n/a 288 8
396.4.ba $$\chi_{396}(47, \cdot)$$ n/a 1696 8
396.4.bf $$\chi_{396}(7, \cdot)$$ n/a 1696 8

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(396))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(396)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(44))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(66))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(99))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(132))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(198))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(396))$$$$^{\oplus 1}$$