## Defining parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$33$$ Sturm bound: $$7680$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(231))$$.

Total New Old
Modular forms 2160 1447 713
Cusp forms 1681 1271 410
Eisenstein series 479 176 303

## Trace form

 $$1271q + 9q^{2} - 13q^{3} - 23q^{4} + 6q^{5} - 33q^{6} - 49q^{7} - 31q^{8} - 45q^{9} + O(q^{10})$$ $$1271q + 9q^{2} - 13q^{3} - 23q^{4} + 6q^{5} - 33q^{6} - 49q^{7} - 31q^{8} - 45q^{9} - 94q^{10} - 15q^{11} - 79q^{12} - 46q^{13} - 33q^{14} - 72q^{15} - 127q^{16} - 30q^{17} - 41q^{18} - 68q^{19} - 46q^{20} - 43q^{21} - 175q^{22} - 16q^{23} - 61q^{24} - 99q^{25} - 58q^{26} + 5q^{27} - 127q^{28} - 22q^{29} - 24q^{30} - 60q^{31} + 17q^{32} + 5q^{33} - 118q^{34} - 34q^{35} - 35q^{36} - 86q^{37} - 28q^{38} - 56q^{39} - 158q^{40} - 42q^{41} - 39q^{42} - 188q^{43} - 143q^{44} - 124q^{45} - 228q^{46} - 104q^{47} - 121q^{48} - 157q^{49} - 121q^{50} - 100q^{51} - 190q^{52} - 74q^{53} - 41q^{54} - 134q^{55} - 99q^{56} - 68q^{57} - 118q^{58} - 48q^{59} + 140q^{60} - 18q^{61} + 24q^{62} + 51q^{63} + 53q^{64} + 132q^{65} + 263q^{66} + 12q^{67} + 274q^{68} + 134q^{69} + 214q^{70} + 216q^{71} + 359q^{72} + 94q^{73} + 198q^{74} + 191q^{75} + 380q^{76} + 113q^{77} + 70q^{78} - 36q^{79} + 378q^{80} + 219q^{81} + 70q^{82} + 4q^{83} + 283q^{84} - 36q^{85} + 224q^{86} + 42q^{87} + 153q^{88} + 110q^{89} + 120q^{90} + 22q^{91} + 68q^{92} - 10q^{93} - 100q^{94} - 108q^{95} + 147q^{96} - 194q^{97} - 223q^{98} - 99q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(231))$$

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
231.2.a $$\chi_{231}(1, \cdot)$$ 231.2.a.a 1 1
231.2.a.b 2
231.2.a.c 2
231.2.a.d 3
231.2.a.e 3
231.2.c $$\chi_{231}(76, \cdot)$$ 231.2.c.a 16 1
231.2.e $$\chi_{231}(188, \cdot)$$ 231.2.e.a 28 1
231.2.g $$\chi_{231}(197, \cdot)$$ 231.2.g.a 24 1
231.2.i $$\chi_{231}(67, \cdot)$$ 231.2.i.a 2 2
231.2.i.b 2
231.2.i.c 2
231.2.i.d 4
231.2.i.e 8
231.2.i.f 10
231.2.j $$\chi_{231}(64, \cdot)$$ 231.2.j.a 4 4
231.2.j.b 4
231.2.j.c 4
231.2.j.d 4
231.2.j.e 4
231.2.j.f 8
231.2.j.g 20
231.2.l $$\chi_{231}(32, \cdot)$$ 231.2.l.a 8 2
231.2.l.b 48
231.2.n $$\chi_{231}(89, \cdot)$$ 231.2.n.a 52 2
231.2.p $$\chi_{231}(10, \cdot)$$ 231.2.p.a 32 2
231.2.s $$\chi_{231}(8, \cdot)$$ 231.2.s.a 96 4
231.2.u $$\chi_{231}(20, \cdot)$$ 231.2.u.a 112 4
231.2.w $$\chi_{231}(13, \cdot)$$ 231.2.w.a 64 4
231.2.y $$\chi_{231}(4, \cdot)$$ 231.2.y.a 64 8
231.2.y.b 64
231.2.ba $$\chi_{231}(19, \cdot)$$ 231.2.ba.a 128 8
231.2.bc $$\chi_{231}(5, \cdot)$$ 231.2.bc.a 224 8
231.2.be $$\chi_{231}(2, \cdot)$$ 231.2.be.a 224 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(231))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(231)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 2}$$