Properties

Label 42.7
Level 42
Weight 7
Dimension 68
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 672
Trace bound 1

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(672\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(42))\).

Total New Old
Modular forms 312 68 244
Cusp forms 264 68 196
Eisenstein series 48 0 48

Trace form

\( 68 q - 84 q^{3} + 128 q^{4} + 672 q^{5} + 384 q^{6} - 1924 q^{7} - 2460 q^{9} + O(q^{10}) \) \( 68 q - 84 q^{3} + 128 q^{4} + 672 q^{5} + 384 q^{6} - 1924 q^{7} - 2460 q^{9} + 192 q^{10} + 11592 q^{11} + 2688 q^{12} + 1720 q^{13} - 10344 q^{15} - 4096 q^{16} - 1680 q^{17} - 672 q^{18} + 14752 q^{19} + 15312 q^{21} - 768 q^{22} + 17808 q^{23} - 1536 q^{24} + 124772 q^{25} - 51072 q^{26} - 141876 q^{27} - 27520 q^{28} + 29760 q^{29} + 35952 q^{30} - 32456 q^{31} + 202740 q^{33} + 101376 q^{34} - 13152 q^{35} - 246912 q^{36} - 269120 q^{37} - 311136 q^{38} - 346332 q^{39} - 6144 q^{40} + 235680 q^{42} + 943768 q^{43} + 370944 q^{44} + 105924 q^{45} + 88608 q^{46} - 378336 q^{47} - 86016 q^{48} - 173740 q^{49} - 644736 q^{50} - 155268 q^{51} - 135680 q^{52} - 267960 q^{53} + 538560 q^{54} + 289656 q^{55} + 79872 q^{56} - 1008672 q^{57} - 159744 q^{58} + 158928 q^{59} + 231552 q^{60} + 1148896 q^{61} - 33384 q^{63} - 655360 q^{64} + 21000 q^{65} + 57792 q^{66} + 847864 q^{67} + 53760 q^{68} + 866088 q^{69} + 1526304 q^{70} + 1403376 q^{71} + 21504 q^{72} + 1317304 q^{73} + 1920576 q^{74} - 1467060 q^{75} - 832256 q^{76} - 3924000 q^{77} - 2053248 q^{78} - 2049704 q^{79} - 688128 q^{80} - 775464 q^{81} - 512448 q^{82} + 512640 q^{84} - 2211840 q^{85} + 2422560 q^{86} + 1250244 q^{87} + 992256 q^{88} + 4544064 q^{89} + 4683648 q^{90} + 4089160 q^{91} - 1148928 q^{92} + 891660 q^{93} - 2995680 q^{94} - 6522432 q^{95} + 49152 q^{96} - 5256800 q^{97} - 1501440 q^{98} - 5012448 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.7.b \(\chi_{42}(29, \cdot)\) 42.7.b.a 12 1
42.7.c \(\chi_{42}(13, \cdot)\) 42.7.c.a 8 1
42.7.g \(\chi_{42}(19, \cdot)\) 42.7.g.a 8 2
42.7.g.b 8
42.7.h \(\chi_{42}(11, \cdot)\) 42.7.h.a 32 2

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

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