## Defining parameters

 Level: $$N$$ = $$261 = 3^{2} \cdot 29$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$17$$ Sturm bound: $$15120$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(261))$$.

Total New Old
Modular forms 5264 4104 1160
Cusp forms 4816 3860 956
Eisenstein series 448 244 204

## Trace form

 $$3860q - 36q^{2} - 50q^{3} - 40q^{4} - 54q^{5} - 74q^{6} - 38q^{7} - 42q^{8} - 38q^{9} + O(q^{10})$$ $$3860q - 36q^{2} - 50q^{3} - 40q^{4} - 54q^{5} - 74q^{6} - 38q^{7} - 42q^{8} - 38q^{9} - 102q^{10} - 36q^{11} - 68q^{12} - 50q^{13} - 54q^{14} - 56q^{15} - 64q^{16} - 42q^{17} - 56q^{18} - 170q^{19} + 138q^{20} - 44q^{21} + 120q^{22} + 124q^{23} + 34q^{24} + 68q^{25} + 28q^{26} - 164q^{27} - 120q^{28} - 148q^{29} - 148q^{30} - 62q^{31} - 320q^{32} - 74q^{33} - 198q^{34} - 238q^{35} - 38q^{36} - 116q^{37} - 256q^{38} - 8q^{39} - 354q^{40} - 56q^{42} - 164q^{43} + 140q^{44} + 52q^{45} + 452q^{46} + 406q^{47} - 122q^{48} + 316q^{49} + 566q^{50} - 218q^{51} + 510q^{52} + 210q^{53} + 106q^{54} + 122q^{55} - 52q^{56} + 10q^{57} - 132q^{58} - 314q^{59} - 92q^{60} - 154q^{61} - 532q^{62} - 128q^{63} - 724q^{64} - 594q^{65} - 20q^{66} - 664q^{67} - 1066q^{68} - 56q^{69} - 2866q^{70} - 2324q^{71} - 2678q^{72} - 2038q^{73} - 3508q^{74} - 1518q^{75} - 3380q^{76} - 1944q^{77} - 1920q^{78} - 554q^{79} - 3570q^{80} - 1014q^{81} - 882q^{82} - 378q^{83} - 716q^{84} - 186q^{85} + 282q^{86} + 318q^{87} + 894q^{88} + 588q^{89} + 956q^{90} + 1166q^{91} + 2886q^{92} + 344q^{93} + 1134q^{94} + 2850q^{95} + 3864q^{96} + 1534q^{97} + 5180q^{98} + 1764q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(261))$$

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
261.3.b $$\chi_{261}(233, \cdot)$$ 261.3.b.a 20 1
261.3.d $$\chi_{261}(260, \cdot)$$ 261.3.d.a 20 1
261.3.f $$\chi_{261}(46, \cdot)$$ 261.3.f.a 8 2
261.3.f.b 8
261.3.f.c 12
261.3.f.d 20
261.3.h $$\chi_{261}(86, \cdot)$$ 261.3.h.a 116 2
261.3.j $$\chi_{261}(59, \cdot)$$ 261.3.j.a 112 2
261.3.m $$\chi_{261}(70, \cdot)$$ 261.3.m.a 232 4
261.3.n $$\chi_{261}(35, \cdot)$$ 261.3.n.a 120 6
261.3.p $$\chi_{261}(53, \cdot)$$ 261.3.p.a 120 6
261.3.s $$\chi_{261}(10, \cdot)$$ 261.3.s.a 48 12
261.3.s.b 120
261.3.s.c 120
261.3.t $$\chi_{261}(20, \cdot)$$ 261.3.t.a 696 12
261.3.v $$\chi_{261}(5, \cdot)$$ 261.3.v.a 696 12
261.3.w $$\chi_{261}(31, \cdot)$$ 261.3.w.a 1392 24

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(261))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(261)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(29))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(87))$$$$^{\oplus 2}$$