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Properties

Label 42.16
Level 42
Weight 16
Dimension 174
Nonzero newspaces 4
Newform subspaces 15
Sturm bound 1536
Trace bound 3

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 16 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 15 \)
Sturm bound: \(1536\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_1(42))\).

Total New Old
Modular forms 744 174 570
Cusp forms 696 174 522
Eisenstein series 48 0 48

Trace form

\( 174 q - 256 q^{2} - 4374 q^{3} - 98304 q^{4} - 230724 q^{5} + 1119744 q^{6} - 12611928 q^{7} - 4194304 q^{8} - 30609024 q^{9} + O(q^{10}) \) \( 174 q - 256 q^{2} - 4374 q^{3} - 98304 q^{4} - 230724 q^{5} + 1119744 q^{6} - 12611928 q^{7} - 4194304 q^{8} - 30609024 q^{9} - 173703168 q^{10} + 207057924 q^{11} - 71663616 q^{12} + 777467016 q^{13} + 550077440 q^{14} - 2515563072 q^{15} - 1610612736 q^{16} + 9645130464 q^{17} + 5654786304 q^{18} + 4600450152 q^{19} + 11522998272 q^{20} - 24653755752 q^{21} + 42418068480 q^{22} + 15868070244 q^{23} - 39309017088 q^{24} - 205249394622 q^{25} + 57062053888 q^{26} + 41841412812 q^{27} - 178273910784 q^{28} - 303640984836 q^{29} + 406087618560 q^{30} + 1024739424096 q^{31} - 68719476736 q^{32} - 336677522670 q^{33} + 1323505015296 q^{34} + 1067723038332 q^{35} + 3460969365504 q^{36} - 3978511881264 q^{37} + 1375550990848 q^{38} + 8180966889420 q^{39} - 2845952704512 q^{40} - 5434024897116 q^{41} - 4266398474496 q^{42} + 19108599329064 q^{43} + 3392437026816 q^{44} + 596227035690 q^{45} - 19430875815936 q^{46} + 6500753628792 q^{47} + 2348273369088 q^{48} + 26621100826422 q^{49} + 8167549935872 q^{50} - 46307603831742 q^{51} - 17961134850048 q^{52} - 16289720319384 q^{53} + 17525049426432 q^{54} + 109541907628308 q^{55} - 4938079928320 q^{56} - 23708735791872 q^{57} - 2933391651840 q^{58} - 3855116019744 q^{59} + 1979968684032 q^{60} - 170660128095636 q^{61} - 10253186788352 q^{62} - 81268344637698 q^{63} - 290271069732864 q^{64} - 179481959343528 q^{65} - 178425043651584 q^{66} - 64895019190728 q^{67} + 158025817522176 q^{68} + 243827800812072 q^{69} - 32584291559424 q^{70} - 174357956213640 q^{71} - 132770470821888 q^{72} - 598286304288588 q^{73} + 477268855487488 q^{74} - 198722626039044 q^{75} + 151391068815360 q^{76} + 1109108478668820 q^{77} - 108658116374016 q^{78} - 2489627242006752 q^{79} - 61934502150144 q^{80} - 1113895641346848 q^{81} + 1289333521476096 q^{82} + 2924181654968856 q^{83} + 408436067991552 q^{84} - 3744843983275872 q^{85} - 998632912275968 q^{86} - 1372419377970264 q^{87} + 342231374364672 q^{88} + 3007360475254236 q^{89} + 234387214811136 q^{90} + 2983342727430120 q^{91} - 460482512683008 q^{92} - 2336754743702118 q^{93} - 3440503655420928 q^{94} + 1862832236075016 q^{95} - 644038935969792 q^{96} + 7396176677118096 q^{97} + 2762794714653440 q^{98} + 6837683289251436 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.16.a \(\chi_{42}(1, \cdot)\) 42.16.a.a 1 1
42.16.a.b 1
42.16.a.c 1
42.16.a.d 1
42.16.a.e 2
42.16.a.f 2
42.16.a.g 2
42.16.a.h 2
42.16.a.i 2
42.16.d \(\chi_{42}(41, \cdot)\) 42.16.d.a 40 1
42.16.e \(\chi_{42}(25, \cdot)\) 42.16.e.a 8 2
42.16.e.b 10
42.16.e.c 10
42.16.e.d 12
42.16.f \(\chi_{42}(5, \cdot)\) 42.16.f.a 80 2

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 1}\)