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Properties

Label 42.12
Level 42
Weight 12
Dimension 126
Nonzero newspaces 4
Newform subspaces 14
Sturm bound 1152
Trace bound 4

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 14 \)
Sturm bound: \(1152\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(42))\).

Total New Old
Modular forms 552 126 426
Cusp forms 504 126 378
Eisenstein series 48 0 48

Trace form

\( 126 q + 64 q^{2} - 2048 q^{4} - 25704 q^{5} + 31104 q^{6} - 191600 q^{7} + 65536 q^{8} - 318030 q^{9} + O(q^{10}) \) \( 126 q + 64 q^{2} - 2048 q^{4} - 25704 q^{5} + 31104 q^{6} - 191600 q^{7} + 65536 q^{8} - 318030 q^{9} - 123648 q^{10} - 1515888 q^{11} + 1201996 q^{13} - 335360 q^{14} - 3967092 q^{15} - 6291456 q^{16} - 2460480 q^{17} + 23977920 q^{18} - 9545720 q^{19} + 4780032 q^{20} + 26593446 q^{21} - 11623680 q^{22} + 3774564 q^{23} - 11403264 q^{24} + 113673438 q^{25} + 129449216 q^{26} + 178999296 q^{28} + 172185444 q^{29} - 949906560 q^{30} - 718340792 q^{31} + 67108864 q^{32} + 1752498216 q^{33} + 579912576 q^{34} - 254443248 q^{35} + 2100627456 q^{36} + 2980087700 q^{37} - 822336640 q^{38} - 549131880 q^{39} - 126615552 q^{40} + 4821581556 q^{41} + 1292644800 q^{42} - 5178130360 q^{43} - 1552269312 q^{44} - 973314720 q^{45} + 1180736256 q^{46} + 4236247164 q^{47} + 5344661994 q^{49} + 7115856832 q^{50} + 10171331496 q^{51} + 1411158016 q^{52} - 5146340256 q^{53} - 10759328640 q^{54} + 16250628348 q^{55} + 7451312128 q^{56} + 31140024276 q^{57} + 24213871104 q^{58} - 27171119160 q^{59} - 2160672768 q^{60} + 14305674964 q^{61} + 13034266880 q^{62} - 48413312040 q^{63} - 53687091200 q^{64} - 20497382868 q^{65} + 58691434752 q^{66} + 64304034672 q^{67} - 2519531520 q^{68} - 12681794808 q^{69} - 60354480384 q^{70} - 26178172344 q^{71} - 16813719552 q^{72} - 216424907528 q^{73} + 22378889216 q^{74} + 128608095096 q^{75} + 4194525184 q^{76} + 18416341308 q^{77} - 37906667904 q^{78} + 332486047512 q^{79} - 26952597504 q^{80} - 115975896678 q^{81} + 87889880448 q^{82} - 477078651120 q^{83} - 17187649536 q^{84} - 250895509512 q^{85} + 19561955456 q^{86} + 156207838248 q^{87} + 6091309056 q^{88} + 150006177636 q^{89} + 34601769216 q^{90} - 139296976784 q^{91} + 67291963392 q^{92} - 370849571388 q^{93} - 343892782080 q^{94} - 492139066524 q^{95} - 11676942336 q^{96} - 165507073496 q^{97} + 54426611008 q^{98} + 1044100415448 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.12.a \(\chi_{42}(1, \cdot)\) 42.12.a.a 1 1
42.12.a.b 1
42.12.a.c 1
42.12.a.d 1
42.12.a.e 1
42.12.a.f 1
42.12.a.g 2
42.12.a.h 2
42.12.d \(\chi_{42}(41, \cdot)\) 42.12.d.a 28 1
42.12.e \(\chi_{42}(25, \cdot)\) 42.12.e.a 6 2
42.12.e.b 6
42.12.e.c 8
42.12.e.d 8
42.12.f \(\chi_{42}(5, \cdot)\) 42.12.f.a 60 2

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)