Properties

Label 42.9
Level 42
Weight 9
Dimension 92
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 864
Trace bound 1

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(864\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(42))\).

Total New Old
Modular forms 408 92 316
Cusp forms 360 92 268
Eisenstein series 48 0 48

Trace form

\( 92 q + 90 q^{3} + 1024 q^{4} - 3348 q^{5} + 2304 q^{6} + 9196 q^{7} - 2298 q^{9} + O(q^{10}) \) \( 92 q + 90 q^{3} + 1024 q^{4} - 3348 q^{5} + 2304 q^{6} + 9196 q^{7} - 2298 q^{9} - 9216 q^{10} + 6516 q^{11} - 11520 q^{12} - 70404 q^{13} + 131028 q^{15} - 65536 q^{16} - 219456 q^{17} - 1536 q^{18} - 717672 q^{19} + 242574 q^{21} + 1015296 q^{22} + 227520 q^{23} - 196608 q^{24} - 2154424 q^{25} - 1391616 q^{26} + 846216 q^{27} + 2395136 q^{28} + 2348280 q^{29} - 2555904 q^{30} - 5728608 q^{31} - 1220466 q^{33} - 3004416 q^{34} + 4350060 q^{35} - 4936704 q^{36} + 16065116 q^{37} - 2557440 q^{38} - 1909824 q^{39} + 1179648 q^{40} - 1569024 q^{42} - 4047112 q^{43} + 834048 q^{44} - 15667602 q^{45} - 8577024 q^{46} - 23565492 q^{47} + 4128768 q^{48} + 30615152 q^{49} + 38854656 q^{50} + 29644362 q^{51} - 6031872 q^{52} - 2550168 q^{53} - 3223296 q^{54} - 24243264 q^{55} - 18284544 q^{56} - 2889180 q^{57} - 6128640 q^{58} + 34182936 q^{59} + 22530816 q^{60} + 187575120 q^{61} - 115354902 q^{63} - 58720256 q^{64} - 71980020 q^{65} - 57748992 q^{66} - 35639264 q^{67} + 28090368 q^{68} + 129898980 q^{69} + 101627904 q^{70} - 41920272 q^{71} + 196608 q^{72} + 117767028 q^{73} - 52862976 q^{74} - 252128748 q^{75} - 92127744 q^{76} - 115448112 q^{77} + 28643328 q^{78} + 207968200 q^{79} + 54853632 q^{80} + 297355854 q^{81} + 231146496 q^{82} - 114563328 q^{84} - 308236488 q^{85} - 391970304 q^{86} - 550922256 q^{87} - 196608000 q^{88} - 113541192 q^{89} + 207386112 q^{90} + 467424552 q^{91} + 133678080 q^{92} + 22441482 q^{93} + 308238336 q^{94} - 368430084 q^{95} + 25165824 q^{96} + 674070624 q^{97} - 45213696 q^{98} + 267064644 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.9.b \(\chi_{42}(29, \cdot)\) 42.9.b.a 16 1
42.9.c \(\chi_{42}(13, \cdot)\) 42.9.c.a 12 1
42.9.g \(\chi_{42}(19, \cdot)\) 42.9.g.a 8 2
42.9.g.b 12
42.9.h \(\chi_{42}(11, \cdot)\) 42.9.h.a 44 2

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)