↑

Properties

Label 42.8
Level 42
Weight 8
Dimension 82
Nonzero newspaces 4
Newform subspaces 13
Sturm bound 768
Trace bound 4

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 13 \)
Sturm bound: \(768\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(42))\).

Total New Old
Modular forms 360 82 278
Cusp forms 312 82 230
Eisenstein series 48 0 48

Trace form

\( 82 q - 16 q^{2} - 384 q^{4} + 216 q^{5} + 864 q^{6} + 2904 q^{7} - 1024 q^{8} - 654 q^{9} + O(q^{10}) \) \( 82 q - 16 q^{2} - 384 q^{4} + 216 q^{5} + 864 q^{6} + 2904 q^{7} - 1024 q^{8} - 654 q^{9} + 3072 q^{10} - 576 q^{11} - 2324 q^{13} + 12608 q^{14} - 15012 q^{15} - 8192 q^{16} - 48792 q^{17} - 4848 q^{18} - 4712 q^{19} + 14592 q^{20} - 24414 q^{21} - 117312 q^{22} - 242172 q^{23} + 61440 q^{24} + 167698 q^{25} + 400768 q^{26} - 50432 q^{28} - 559044 q^{29} - 324480 q^{30} - 743048 q^{31} - 65536 q^{32} - 79020 q^{33} + 1048992 q^{34} + 1431072 q^{35} + 1433472 q^{36} - 34140 q^{37} - 277088 q^{38} - 1579440 q^{39} + 196608 q^{40} - 1157220 q^{41} - 731664 q^{42} - 2674776 q^{43} - 36864 q^{44} + 4694220 q^{45} + 1771776 q^{46} + 33900 q^{47} + 4325398 q^{49} + 4350992 q^{50} - 5261184 q^{51} - 1157888 q^{52} - 3483864 q^{53} + 1818144 q^{54} - 6930612 q^{55} - 919552 q^{56} + 5421324 q^{57} + 4583616 q^{58} + 5529384 q^{59} + 18432 q^{60} - 16472564 q^{61} + 1802560 q^{62} - 3826848 q^{63} - 7864320 q^{64} + 4126692 q^{65} - 2586816 q^{66} + 19539824 q^{67} - 3122688 q^{68} + 5646024 q^{69} + 675456 q^{70} - 14826744 q^{71} - 1182720 q^{72} - 15579272 q^{73} + 877504 q^{74} + 9886536 q^{75} + 4945408 q^{76} + 30622956 q^{77} + 14567712 q^{78} + 1098152 q^{79} + 884736 q^{80} - 12114846 q^{81} - 29210976 q^{82} - 3440976 q^{83} - 11252352 q^{84} - 24310152 q^{85} - 14784416 q^{86} - 17714880 q^{87} - 5965824 q^{88} + 14311164 q^{89} + 10077696 q^{90} + 62670208 q^{91} + 8683008 q^{92} + 45078768 q^{93} + 612288 q^{94} - 19549164 q^{95} + 3932160 q^{96} - 24261008 q^{97} - 30737872 q^{98} - 17633160 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.8.a \(\chi_{42}(1, \cdot)\) 42.8.a.a 1 1
42.8.a.b 1
42.8.a.c 1
42.8.a.d 1
42.8.a.e 1
42.8.a.f 1
42.8.d \(\chi_{42}(41, \cdot)\) 42.8.d.a 20 1
42.8.e \(\chi_{42}(25, \cdot)\) 42.8.e.a 2 2
42.8.e.b 2
42.8.e.c 4
42.8.e.d 6
42.8.e.e 6
42.8.f \(\chi_{42}(5, \cdot)\) 42.8.f.a 36 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

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