## Defining parameters

 Level: $$N$$ = $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$8$$ Newform subspaces: $$29$$ Sturm bound: $$14112$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(294))$$.

Total New Old
Modular forms 4944 1128 3816
Cusp forms 4464 1128 3336
Eisenstein series 480 0 480

## Trace form

 $$1128q - 12q^{3} - 16q^{4} - 24q^{5} + 16q^{7} - 36q^{9} + O(q^{10})$$ $$1128q - 12q^{3} - 16q^{4} - 24q^{5} + 16q^{7} - 36q^{9} + 96q^{10} + 72q^{11} + 24q^{12} + 56q^{13} + 168q^{15} + 96q^{17} + 48q^{18} + 80q^{19} + 18q^{21} - 96q^{23} - 48q^{24} - 264q^{25} - 192q^{26} - 504q^{27} - 72q^{28} - 240q^{29} - 264q^{30} - 16q^{31} + 204q^{33} + 132q^{35} + 144q^{36} + 1176q^{37} + 792q^{38} + 1050q^{39} + 480q^{40} + 1176q^{41} + 576q^{42} + 712q^{43} + 480q^{44} + 264q^{45} + 648q^{46} + 432q^{47} - 204q^{49} - 144q^{50} - 324q^{51} + 160q^{52} - 648q^{53} + 288q^{54} - 1932q^{55} - 576q^{56} - 252q^{57} - 912q^{58} - 1800q^{59} - 288q^{60} - 2516q^{61} - 1176q^{62} - 492q^{63} + 128q^{64} - 840q^{65} - 672q^{66} - 1104q^{67} - 192q^{68} - 504q^{69} - 504q^{70} - 384q^{71} - 96q^{72} - 760q^{73} - 480q^{74} - 720q^{75} - 784q^{76} - 144q^{77} - 576q^{78} - 288q^{79} + 96q^{80} + 1428q^{81} - 480q^{82} + 2352q^{83} + 12q^{84} + 1776q^{85} + 528q^{86} + 2328q^{87} + 288q^{88} + 1488q^{89} + 1344q^{90} + 1456q^{91} + 96q^{92} + 2100q^{93} + 912q^{94} + 1416q^{95} + 96q^{96} + 896q^{97} + 192q^{98} - 504q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(294))$$

We only show spaces with odd 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
294.3.b $$\chi_{294}(197, \cdot)$$ 294.3.b.a 2 1
294.3.b.b 2
294.3.b.c 2
294.3.b.d 2
294.3.b.e 4
294.3.b.f 4
294.3.b.g 4
294.3.b.h 4
294.3.b.i 4
294.3.c $$\chi_{294}(97, \cdot)$$ 294.3.c.a 4 1
294.3.c.b 8
294.3.g $$\chi_{294}(19, \cdot)$$ 294.3.g.a 4 2
294.3.g.b 4
294.3.g.c 4
294.3.g.d 8
294.3.g.e 8
294.3.h $$\chi_{294}(263, \cdot)$$ 294.3.h.a 4 2
294.3.h.b 4
294.3.h.c 4
294.3.h.d 8
294.3.h.e 8
294.3.h.f 8
294.3.h.g 8
294.3.h.h 8
294.3.k $$\chi_{294}(13, \cdot)$$ 294.3.k.a 120 6
294.3.l $$\chi_{294}(29, \cdot)$$ 294.3.l.a 216 6
294.3.n $$\chi_{294}(11, \cdot)$$ 294.3.n.a 456 12
294.3.o $$\chi_{294}(61, \cdot)$$ 294.3.o.a 96 12
294.3.o.b 120

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(294))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(294)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 2}$$