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Properties

Label 77.8
Level 77
Weight 8
Dimension 1538
Nonzero newspaces 8
Newform subspaces 15
Sturm bound 3840
Trace bound 2

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Defining parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 15 \)
Sturm bound: \(3840\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(77))\).

Total New Old
Modular forms 1740 1630 110
Cusp forms 1620 1538 82
Eisenstein series 120 92 28

Trace form

\( 1538 q - 14 q^{2} - 68 q^{3} + 498 q^{4} - 8 q^{5} - 6040 q^{6} + 302 q^{7} + 12576 q^{8} - 4446 q^{9} + O(q^{10}) \) \( 1538 q - 14 q^{2} - 68 q^{3} + 498 q^{4} - 8 q^{5} - 6040 q^{6} + 302 q^{7} + 12576 q^{8} - 4446 q^{9} - 1830 q^{10} - 7858 q^{11} - 7500 q^{12} + 33850 q^{13} - 48196 q^{14} - 113082 q^{15} + 58802 q^{16} + 183166 q^{17} + 467060 q^{18} - 82414 q^{19} - 1034314 q^{20} - 468992 q^{21} - 244634 q^{22} + 652742 q^{23} + 782484 q^{24} + 495006 q^{25} + 584722 q^{26} - 144392 q^{27} - 777478 q^{28} - 624924 q^{29} - 117098 q^{30} + 738012 q^{31} - 296068 q^{32} - 2697612 q^{33} - 3245836 q^{34} - 93724 q^{35} + 4666666 q^{36} + 2698956 q^{37} + 6461298 q^{38} + 5973666 q^{39} - 310526 q^{40} - 5963678 q^{41} - 5384316 q^{42} - 6596880 q^{43} - 11845914 q^{44} - 7232816 q^{45} + 3028962 q^{46} + 11214626 q^{47} + 32524846 q^{48} + 11409764 q^{49} + 8146880 q^{50} + 7769566 q^{51} - 15998298 q^{52} - 24630018 q^{53} - 50414046 q^{54} - 15071212 q^{55} + 5303526 q^{56} + 7405608 q^{57} + 20895790 q^{58} + 16425920 q^{59} + 32483330 q^{60} + 7774452 q^{61} + 19687582 q^{62} + 41196954 q^{63} + 5899236 q^{64} - 19958912 q^{65} - 59116676 q^{66} - 43886394 q^{67} - 66160654 q^{68} - 58801174 q^{69} - 47572294 q^{70} + 26462898 q^{71} + 51202198 q^{72} - 3630168 q^{73} + 73314270 q^{74} + 56499970 q^{75} + 76560084 q^{76} + 34072470 q^{77} + 78560720 q^{78} + 58121908 q^{79} + 52028714 q^{80} - 6296880 q^{81} - 40599388 q^{82} - 92662860 q^{83} - 315298046 q^{84} - 167579680 q^{85} - 81800148 q^{86} - 37530060 q^{87} - 27110196 q^{88} + 30411714 q^{89} + 253849902 q^{90} + 134306576 q^{91} + 212308840 q^{92} + 99478326 q^{93} - 17387902 q^{94} - 55950324 q^{95} - 179757498 q^{96} - 2057540 q^{97} - 58638140 q^{98} + 19140106 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(77))\)

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
77.8.a \(\chi_{77}(1, \cdot)\) 77.8.a.a 7 1
77.8.a.b 9
77.8.a.c 9
77.8.a.d 11
77.8.b \(\chi_{77}(76, \cdot)\) 77.8.b.a 2 1
77.8.b.b 52
77.8.e \(\chi_{77}(23, \cdot)\) 77.8.e.a 46 2
77.8.e.b 46
77.8.f \(\chi_{77}(15, \cdot)\) 77.8.f.a 80 4
77.8.f.b 88
77.8.i \(\chi_{77}(10, \cdot)\) 77.8.i.a 108 2
77.8.l \(\chi_{77}(6, \cdot)\) 77.8.l.a 8 4
77.8.l.b 208
77.8.m \(\chi_{77}(4, \cdot)\) 77.8.m.a 432 8
77.8.n \(\chi_{77}(17, \cdot)\) 77.8.n.a 432 8

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(77))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(77)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)