## Defining parameters

 Level: $$N$$ = $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$32$$ Sturm bound: $$1204224$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3136))$$.

Total New Old
Modular forms 305376 163571 141805
Cusp forms 296737 161437 135300
Eisenstein series 8639 2134 6505

## Trace form

 $$161437q - 248q^{2} - 186q^{3} - 248q^{4} - 248q^{5} - 248q^{6} - 216q^{7} - 440q^{8} - 313q^{9} + O(q^{10})$$ $$161437q - 248q^{2} - 186q^{3} - 248q^{4} - 248q^{5} - 248q^{6} - 216q^{7} - 440q^{8} - 313q^{9} - 248q^{10} - 190q^{11} - 248q^{12} - 256q^{13} - 288q^{14} - 336q^{15} - 248q^{16} - 442q^{17} - 248q^{18} - 194q^{19} - 248q^{20} - 288q^{21} - 432q^{22} - 188q^{23} - 208q^{24} - 299q^{25} - 208q^{26} - 180q^{27} - 288q^{28} - 424q^{29} - 168q^{30} - 164q^{31} - 208q^{32} - 160q^{33} - 208q^{34} - 216q^{35} - 360q^{36} - 240q^{37} - 208q^{38} - 188q^{39} - 208q^{40} - 310q^{41} - 288q^{42} - 342q^{43} - 240q^{44} - 252q^{45} - 248q^{46} - 204q^{47} - 248q^{48} - 504q^{49} - 800q^{50} - 160q^{51} - 296q^{52} - 272q^{53} - 312q^{54} - 100q^{55} - 288q^{56} - 556q^{57} - 320q^{58} - 118q^{59} - 344q^{60} - 256q^{61} - 280q^{62} - 216q^{63} - 536q^{64} - 684q^{65} - 328q^{66} - 98q^{67} - 296q^{68} - 260q^{69} - 288q^{70} - 268q^{71} - 320q^{72} - 310q^{73} - 304q^{74} - 130q^{75} - 312q^{76} - 288q^{77} - 464q^{78} - 156q^{79} - 208q^{80} - 359q^{81} - 168q^{82} - 186q^{83} - 288q^{84} - 432q^{85} - 144q^{86} - 188q^{87} - 168q^{88} - 262q^{89} - 104q^{90} - 132q^{91} - 288q^{92} - 112q^{93} - 152q^{94} - 24q^{95} - 112q^{96} + 46q^{97} - 288q^{98} - 318q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3136))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
3136.2.a $$\chi_{3136}(1, \cdot)$$ 3136.2.a.a 1 1
3136.2.a.b 1
3136.2.a.c 1
3136.2.a.d 1
3136.2.a.e 1
3136.2.a.f 1
3136.2.a.g 1
3136.2.a.h 1
3136.2.a.i 1
3136.2.a.j 1
3136.2.a.k 1
3136.2.a.l 1
3136.2.a.m 1
3136.2.a.n 1
3136.2.a.o 1
3136.2.a.p 1
3136.2.a.q 1
3136.2.a.r 1
3136.2.a.s 1
3136.2.a.t 1
3136.2.a.u 1
3136.2.a.v 1
3136.2.a.w 1
3136.2.a.x 1
3136.2.a.y 1
3136.2.a.z 1
3136.2.a.ba 1
3136.2.a.bb 1
3136.2.a.bc 1
3136.2.a.bd 2
3136.2.a.be 2
3136.2.a.bf 2
3136.2.a.bg 2
3136.2.a.bh 2
3136.2.a.bi 2
3136.2.a.bj 2
3136.2.a.bk 2
3136.2.a.bl 2
3136.2.a.bm 2
3136.2.a.bn 2
3136.2.a.bo 2
3136.2.a.bp 2
3136.2.a.bq 2
3136.2.a.br 2
3136.2.a.bs 2
3136.2.a.bt 2
3136.2.a.bu 2
3136.2.a.bv 2
3136.2.a.bw 2
3136.2.a.bx 2
3136.2.a.by 2
3136.2.a.bz 4
3136.2.b $$\chi_{3136}(1569, \cdot)$$ 3136.2.b.a 2 1
3136.2.b.b 2
3136.2.b.c 2
3136.2.b.d 4
3136.2.b.e 4
3136.2.b.f 4
3136.2.b.g 4
3136.2.b.h 4
3136.2.b.i 4
3136.2.b.j 4
3136.2.b.k 8
3136.2.b.l 12
3136.2.b.m 12
3136.2.b.n 16
3136.2.e $$\chi_{3136}(1567, \cdot)$$ 3136.2.e.a 8 1
3136.2.e.b 8
3136.2.e.c 8
3136.2.e.d 12
3136.2.e.e 12
3136.2.e.f 32
3136.2.f $$\chi_{3136}(3135, \cdot)$$ 3136.2.f.a 2 1
3136.2.f.b 2
3136.2.f.c 4
3136.2.f.d 4
3136.2.f.e 4
3136.2.f.f 4
3136.2.f.g 8
3136.2.f.h 8
3136.2.f.i 8
3136.2.f.j 16
3136.2.f.k 16
3136.2.i $$\chi_{3136}(961, \cdot)$$ n/a 152 2
3136.2.j $$\chi_{3136}(783, \cdot)$$ n/a 152 2
3136.2.m $$\chi_{3136}(785, \cdot)$$ n/a 154 2
3136.2.p $$\chi_{3136}(1599, \cdot)$$ n/a 152 2
3136.2.q $$\chi_{3136}(31, \cdot)$$ n/a 160 2
3136.2.t $$\chi_{3136}(2529, \cdot)$$ n/a 160 2
3136.2.u $$\chi_{3136}(449, \cdot)$$ n/a 660 6
3136.2.v $$\chi_{3136}(393, \cdot)$$ None 0 4
3136.2.y $$\chi_{3136}(391, \cdot)$$ None 0 4
3136.2.ba $$\chi_{3136}(815, \cdot)$$ n/a 304 4
3136.2.bb $$\chi_{3136}(177, \cdot)$$ n/a 304 4
3136.2.bf $$\chi_{3136}(447, \cdot)$$ n/a 660 6
3136.2.bg $$\chi_{3136}(223, \cdot)$$ n/a 672 6
3136.2.bj $$\chi_{3136}(225, \cdot)$$ n/a 672 6
3136.2.bk $$\chi_{3136}(197, \cdot)$$ n/a 2584 8
3136.2.bl $$\chi_{3136}(195, \cdot)$$ n/a 2528 8
3136.2.bo $$\chi_{3136}(65, \cdot)$$ n/a 1320 12
3136.2.bq $$\chi_{3136}(361, \cdot)$$ None 0 8
3136.2.br $$\chi_{3136}(215, \cdot)$$ None 0 8
3136.2.bt $$\chi_{3136}(113, \cdot)$$ n/a 1320 12
3136.2.bw $$\chi_{3136}(111, \cdot)$$ n/a 1320 12
3136.2.bx $$\chi_{3136}(289, \cdot)$$ n/a 1344 12
3136.2.ca $$\chi_{3136}(159, \cdot)$$ n/a 1344 12
3136.2.cb $$\chi_{3136}(255, \cdot)$$ n/a 1320 12
3136.2.cg $$\chi_{3136}(19, \cdot)$$ n/a 5056 16
3136.2.ch $$\chi_{3136}(165, \cdot)$$ n/a 5056 16
3136.2.ci $$\chi_{3136}(55, \cdot)$$ None 0 24
3136.2.cl $$\chi_{3136}(57, \cdot)$$ None 0 24
3136.2.cn $$\chi_{3136}(81, \cdot)$$ n/a 2640 24
3136.2.co $$\chi_{3136}(47, \cdot)$$ n/a 2640 24
3136.2.cq $$\chi_{3136}(27, \cdot)$$ n/a 21408 48
3136.2.cr $$\chi_{3136}(29, \cdot)$$ n/a 21408 48
3136.2.cv $$\chi_{3136}(87, \cdot)$$ None 0 48
3136.2.cw $$\chi_{3136}(9, \cdot)$$ None 0 48
3136.2.da $$\chi_{3136}(37, \cdot)$$ n/a 42816 96
3136.2.db $$\chi_{3136}(3, \cdot)$$ n/a 42816 96

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3136))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3136)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(448))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(784))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1568))$$$$^{\oplus 2}$$