# Properties

 Label 704.4 Level 704 Weight 4 Dimension 25290 Nonzero newspaces 16 Sturm bound 122880 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$122880$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(704))$$.

Total New Old
Modular forms 46800 25686 21114
Cusp forms 45360 25290 20070
Eisenstein series 1440 396 1044

## Trace form

 $$25290 q - 64 q^{2} - 48 q^{3} - 64 q^{4} - 64 q^{5} - 64 q^{6} - 44 q^{7} - 64 q^{8} - 26 q^{9} + O(q^{10})$$ $$25290 q - 64 q^{2} - 48 q^{3} - 64 q^{4} - 64 q^{5} - 64 q^{6} - 44 q^{7} - 64 q^{8} - 26 q^{9} - 64 q^{10} - 34 q^{11} - 144 q^{12} - 208 q^{13} - 64 q^{14} - 292 q^{15} - 64 q^{16} - 320 q^{17} - 64 q^{18} - 96 q^{19} - 64 q^{20} - 40 q^{21} - 544 q^{22} - 104 q^{23} - 2064 q^{24} - 182 q^{25} + 16 q^{26} + 324 q^{27} + 1456 q^{28} + 736 q^{29} + 4576 q^{30} + 676 q^{31} + 2416 q^{32} + 928 q^{33} + 1856 q^{34} + 908 q^{35} + 1696 q^{36} + 976 q^{37} - 944 q^{38} - 44 q^{39} - 3344 q^{40} - 2128 q^{41} - 6384 q^{42} - 1720 q^{43} - 1072 q^{44} - 3112 q^{45} - 64 q^{46} - 1932 q^{47} - 64 q^{48} - 2238 q^{49} + 5648 q^{50} - 8996 q^{51} + 6560 q^{52} - 880 q^{53} + 3392 q^{54} - 628 q^{55} - 928 q^{56} + 2076 q^{57} - 4816 q^{58} + 8872 q^{59} - 9856 q^{60} + 2096 q^{61} - 6048 q^{62} + 15308 q^{63} - 12160 q^{64} + 4372 q^{65} - 5608 q^{66} + 11940 q^{67} - 4192 q^{68} + 1208 q^{69} - 4096 q^{70} + 852 q^{71} + 1232 q^{72} - 2128 q^{73} + 5200 q^{74} - 14784 q^{75} + 11840 q^{76} - 3196 q^{77} + 3840 q^{78} - 20204 q^{79} - 8592 q^{80} - 13750 q^{81} - 13984 q^{82} - 5168 q^{83} - 8352 q^{84} - 3376 q^{85} + 976 q^{86} - 64 q^{87} + 3048 q^{88} + 6956 q^{89} + 18656 q^{90} + 6620 q^{91} + 25168 q^{92} + 16688 q^{93} + 17792 q^{94} + 13740 q^{95} + 25776 q^{96} + 18656 q^{97} + 24144 q^{98} + 4690 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(704))$$

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
704.4.a $$\chi_{704}(1, \cdot)$$ 704.4.a.a 1 1
704.4.a.b 1
704.4.a.c 1
704.4.a.d 1
704.4.a.e 1
704.4.a.f 1
704.4.a.g 1
704.4.a.h 1
704.4.a.i 1
704.4.a.j 1
704.4.a.k 1
704.4.a.l 1
704.4.a.m 2
704.4.a.n 2
704.4.a.o 2
704.4.a.p 2
704.4.a.q 2
704.4.a.r 2
704.4.a.s 3
704.4.a.t 3
704.4.a.u 3
704.4.a.v 3
704.4.a.w 4
704.4.a.x 4
704.4.a.y 4
704.4.a.z 4
704.4.a.ba 4
704.4.a.bb 4
704.4.c $$\chi_{704}(353, \cdot)$$ 704.4.c.a 4 1
704.4.c.b 4
704.4.c.c 12
704.4.c.d 16
704.4.c.e 24
704.4.e $$\chi_{704}(703, \cdot)$$ 704.4.e.a 2 1
704.4.e.b 2
704.4.e.c 2
704.4.e.d 4
704.4.e.e 8
704.4.e.f 16
704.4.e.g 36
704.4.g $$\chi_{704}(351, \cdot)$$ 704.4.g.a 4 1
704.4.g.b 4
704.4.g.c 8
704.4.g.d 16
704.4.g.e 40
704.4.i $$\chi_{704}(175, \cdot)$$ n/a 140 2
704.4.j $$\chi_{704}(177, \cdot)$$ n/a 120 2
704.4.m $$\chi_{704}(257, \cdot)$$ n/a 280 4
704.4.n $$\chi_{704}(89, \cdot)$$ None 0 4
704.4.q $$\chi_{704}(87, \cdot)$$ None 0 4
704.4.s $$\chi_{704}(95, \cdot)$$ n/a 288 4
704.4.u $$\chi_{704}(63, \cdot)$$ n/a 280 4
704.4.w $$\chi_{704}(97, \cdot)$$ n/a 288 4
704.4.z $$\chi_{704}(45, \cdot)$$ n/a 1920 8
704.4.bb $$\chi_{704}(43, \cdot)$$ n/a 2288 8
704.4.be $$\chi_{704}(49, \cdot)$$ n/a 560 8
704.4.bf $$\chi_{704}(79, \cdot)$$ n/a 560 8
704.4.bg $$\chi_{704}(7, \cdot)$$ None 0 16
704.4.bj $$\chi_{704}(9, \cdot)$$ None 0 16
704.4.bk $$\chi_{704}(19, \cdot)$$ n/a 9152 32
704.4.bm $$\chi_{704}(5, \cdot)$$ n/a 9152 32

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(704))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(704)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(44))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(88))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(176))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(352))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(704))$$$$^{\oplus 1}$$