## Defining parameters

 Level: $$N$$ = $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$150528$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(735))$$.

Total New Old
Modular forms 57408 37534 19874
Cusp forms 55488 36950 18538
Eisenstein series 1920 584 1336

## Trace form

 $$36950 q + 4 q^{2} - 12 q^{3} - 84 q^{4} - 42 q^{5} - 222 q^{6} - 168 q^{7} + 180 q^{8} + 120 q^{9} + O(q^{10})$$ $$36950 q + 4 q^{2} - 12 q^{3} - 84 q^{4} - 42 q^{5} - 222 q^{6} - 168 q^{7} + 180 q^{8} + 120 q^{9} - 78 q^{10} - 56 q^{11} - 294 q^{12} - 208 q^{13} - 624 q^{14} - 291 q^{15} - 356 q^{16} - 8 q^{17} + 858 q^{18} + 1940 q^{19} + 1652 q^{20} + 1068 q^{21} + 1148 q^{22} - 960 q^{23} - 1266 q^{24} - 3532 q^{25} - 4592 q^{26} - 1332 q^{27} - 3048 q^{28} - 1420 q^{29} + 33 q^{30} - 412 q^{31} + 3308 q^{32} + 606 q^{33} + 4772 q^{34} + 2532 q^{35} + 6342 q^{36} - 984 q^{37} + 256 q^{38} + 1770 q^{39} - 2294 q^{40} - 1708 q^{41} + 4038 q^{42} + 4376 q^{43} + 12824 q^{44} + 429 q^{45} + 14828 q^{46} + 5272 q^{47} + 4536 q^{48} + 11376 q^{49} + 2116 q^{50} - 5742 q^{51} + 492 q^{52} - 3584 q^{53} - 16866 q^{54} - 14418 q^{55} - 13500 q^{56} - 10002 q^{57} - 24172 q^{58} - 19136 q^{59} - 8967 q^{60} - 8520 q^{61} - 9336 q^{62} + 2916 q^{63} + 22308 q^{64} + 15808 q^{65} + 21162 q^{66} + 27952 q^{67} + 29384 q^{68} + 6726 q^{69} + 21234 q^{70} + 10208 q^{71} - 9462 q^{72} + 1568 q^{73} + 11576 q^{74} - 2895 q^{75} - 10700 q^{76} + 1512 q^{77} - 7614 q^{78} - 11660 q^{79} - 36850 q^{80} - 12756 q^{81} - 67992 q^{82} - 41496 q^{83} - 34332 q^{84} - 29454 q^{85} - 34508 q^{86} + 4650 q^{87} - 23208 q^{88} + 516 q^{89} + 34896 q^{90} + 19884 q^{91} + 21636 q^{92} + 42906 q^{93} + 69656 q^{94} + 36568 q^{95} + 49464 q^{96} + 36156 q^{97} + 102108 q^{98} + 864 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(735))$$

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
735.4.a $$\chi_{735}(1, \cdot)$$ 735.4.a.a 1 1
735.4.a.b 1
735.4.a.c 1
735.4.a.d 1
735.4.a.e 1
735.4.a.f 1
735.4.a.g 1
735.4.a.h 1
735.4.a.i 1
735.4.a.j 1
735.4.a.k 2
735.4.a.l 2
735.4.a.m 2
735.4.a.n 2
735.4.a.o 2
735.4.a.p 2
735.4.a.q 2
735.4.a.r 3
735.4.a.s 3
735.4.a.t 4
735.4.a.u 4
735.4.a.v 4
735.4.a.w 4
735.4.a.x 5
735.4.a.y 5
735.4.a.z 5
735.4.a.ba 5
735.4.a.bb 8
735.4.a.bc 8
735.4.b $$\chi_{735}(146, \cdot)$$ n/a 160 1
735.4.d $$\chi_{735}(589, \cdot)$$ n/a 124 1
735.4.g $$\chi_{735}(734, \cdot)$$ n/a 232 1
735.4.i $$\chi_{735}(226, \cdot)$$ n/a 160 2
735.4.j $$\chi_{735}(197, \cdot)$$ n/a 472 2
735.4.m $$\chi_{735}(97, \cdot)$$ n/a 240 2
735.4.p $$\chi_{735}(374, \cdot)$$ n/a 464 2
735.4.q $$\chi_{735}(79, \cdot)$$ n/a 240 2
735.4.s $$\chi_{735}(521, \cdot)$$ n/a 320 2
735.4.u $$\chi_{735}(106, \cdot)$$ n/a 672 6
735.4.v $$\chi_{735}(178, \cdot)$$ n/a 480 4
735.4.y $$\chi_{735}(128, \cdot)$$ n/a 928 4
735.4.ba $$\chi_{735}(104, \cdot)$$ n/a 1992 6
735.4.bd $$\chi_{735}(64, \cdot)$$ n/a 1008 6
735.4.bf $$\chi_{735}(41, \cdot)$$ n/a 1344 6
735.4.bg $$\chi_{735}(16, \cdot)$$ n/a 1344 12
735.4.bi $$\chi_{735}(13, \cdot)$$ n/a 2016 12
735.4.bj $$\chi_{735}(8, \cdot)$$ n/a 3984 12
735.4.bm $$\chi_{735}(26, \cdot)$$ n/a 2688 12
735.4.bo $$\chi_{735}(4, \cdot)$$ n/a 2016 12
735.4.bp $$\chi_{735}(59, \cdot)$$ n/a 3984 12
735.4.bt $$\chi_{735}(2, \cdot)$$ n/a 7968 24
735.4.bu $$\chi_{735}(52, \cdot)$$ n/a 4032 24

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(735))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(735)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 2}$$