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Properties

Label 609.2
Level 609
Weight 2
Dimension 9371
Nonzero newspaces 24
Newform subspaces 54
Sturm bound 53760
Trace bound 4

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Defining parameters

Level: \( N \) = \( 609 = 3 \cdot 7 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 24 \)
Newform subspaces: \( 54 \)
Sturm bound: \(53760\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(609))\).

Total New Old
Modular forms 14112 9907 4205
Cusp forms 12769 9371 3398
Eisenstein series 1343 536 807

Trace form

\( 9371 q + 9 q^{2} - 49 q^{3} - 95 q^{4} + 6 q^{5} - 59 q^{6} - 129 q^{7} + 9 q^{8} - 61 q^{9} + O(q^{10}) \) \( 9371 q + 9 q^{2} - 49 q^{3} - 95 q^{4} + 6 q^{5} - 59 q^{6} - 129 q^{7} + 9 q^{8} - 61 q^{9} - 106 q^{10} - 55 q^{12} - 98 q^{13} - 3 q^{14} - 122 q^{15} - 79 q^{16} + 30 q^{17} - 47 q^{18} - 80 q^{19} - 30 q^{20} - 91 q^{21} - 332 q^{22} - 32 q^{23} - 215 q^{24} - 203 q^{25} - 98 q^{26} - 145 q^{27} - 307 q^{28} - 89 q^{29} - 274 q^{30} - 204 q^{31} - 167 q^{32} - 140 q^{33} - 210 q^{34} - 50 q^{35} - 283 q^{36} - 134 q^{37} - 40 q^{38} - 98 q^{39} - 94 q^{40} + 78 q^{41} - 49 q^{42} - 228 q^{43} + 28 q^{44} - 78 q^{45} - 320 q^{46} - 76 q^{47} - 231 q^{48} - 249 q^{49} - 293 q^{50} - 138 q^{51} - 450 q^{52} - 186 q^{53} - 47 q^{54} - 488 q^{55} - 175 q^{56} - 308 q^{57} - 691 q^{58} - 100 q^{59} - 418 q^{60} - 274 q^{61} - 368 q^{62} - 127 q^{63} - 671 q^{64} - 180 q^{65} - 244 q^{66} - 264 q^{67} - 278 q^{68} - 144 q^{69} - 310 q^{70} - 16 q^{71} + 65 q^{72} - 118 q^{73} + 82 q^{74} + 145 q^{75} + 28 q^{76} + 36 q^{77} + 114 q^{78} - 28 q^{79} + 198 q^{80} + 187 q^{81} - 22 q^{82} + 84 q^{83} + 69 q^{84} - 196 q^{85} + 144 q^{86} + 119 q^{87} - 68 q^{88} + 150 q^{89} + 254 q^{90} - 138 q^{91} + 168 q^{92} + 136 q^{93} + 8 q^{94} + 132 q^{95} + 309 q^{96} - 222 q^{97} - 143 q^{98} - 212 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(609))\)

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
609.2.a \(\chi_{609}(1, \cdot)\) 609.2.a.a 1 1
609.2.a.b 1
609.2.a.c 2
609.2.a.d 3
609.2.a.e 3
609.2.a.f 4
609.2.a.g 4
609.2.a.h 4
609.2.a.i 5
609.2.d \(\chi_{609}(146, \cdot)\) 609.2.d.a 76 1
609.2.e \(\chi_{609}(463, \cdot)\) 609.2.e.a 14 1
609.2.e.b 18
609.2.h \(\chi_{609}(608, \cdot)\) 609.2.h.a 8 1
609.2.h.b 8
609.2.h.c 12
609.2.h.d 12
609.2.h.e 12
609.2.h.f 24
609.2.i \(\chi_{609}(88, \cdot)\) 609.2.i.a 2 2
609.2.i.b 2
609.2.i.c 12
609.2.i.d 12
609.2.i.e 24
609.2.i.f 24
609.2.l \(\chi_{609}(302, \cdot)\) 609.2.l.a 60 2
609.2.l.b 60
609.2.m \(\chi_{609}(244, \cdot)\) 609.2.m.a 80 2
609.2.n \(\chi_{609}(173, \cdot)\) 609.2.n.a 8 2
609.2.n.b 12
609.2.n.c 12
609.2.n.d 120
609.2.q \(\chi_{609}(59, \cdot)\) 609.2.q.a 4 2
609.2.q.b 144
609.2.r \(\chi_{609}(289, \cdot)\) 609.2.r.a 80 2
609.2.u \(\chi_{609}(169, \cdot)\) 609.2.u.a 36 6
609.2.u.b 36
609.2.u.c 48
609.2.u.d 48
609.2.v \(\chi_{609}(128, \cdot)\) 609.2.v.a 304 4
609.2.w \(\chi_{609}(157, \cdot)\) 609.2.w.a 160 4
609.2.z \(\chi_{609}(62, \cdot)\) 609.2.z.a 456 6
609.2.bc \(\chi_{609}(22, \cdot)\) 609.2.bc.a 84 6
609.2.bc.b 108
609.2.bd \(\chi_{609}(20, \cdot)\) 609.2.bd.a 456 6
609.2.bg \(\chi_{609}(16, \cdot)\) 609.2.bg.a 240 12
609.2.bg.b 240
609.2.bh \(\chi_{609}(55, \cdot)\) 609.2.bh.a 480 12
609.2.bi \(\chi_{609}(8, \cdot)\) 609.2.bi.a 360 12
609.2.bi.b 360
609.2.bn \(\chi_{609}(4, \cdot)\) 609.2.bn.a 480 12
609.2.bo \(\chi_{609}(110, \cdot)\) 609.2.bo.a 912 12
609.2.br \(\chi_{609}(5, \cdot)\) 609.2.br.a 912 12
609.2.bu \(\chi_{609}(10, \cdot)\) 609.2.bu.a 960 24
609.2.bv \(\chi_{609}(2, \cdot)\) 609.2.bv.a 1824 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(609))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(609)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(87))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(203))\)\(^{\oplus 2}\)