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Properties

Label 471.8
Level 471
Weight 8
Dimension 42976
Nonzero newspaces 12
Sturm bound 131456
Trace bound 1

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Defining parameters

Level: \( N \) = \( 471 = 3 \cdot 157 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(131456\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(471))\).

Total New Old
Modular forms 57824 43288 14536
Cusp forms 57200 42976 14224
Eisenstein series 624 312 312

Trace form

\( 42976 q - 12 q^{2} - 24 q^{3} + 28 q^{4} - 780 q^{5} + 246 q^{6} - 28 q^{7} + 2640 q^{8} - 1536 q^{9} + O(q^{10}) \) \( 42976 q - 12 q^{2} - 24 q^{3} + 28 q^{4} - 780 q^{5} + 246 q^{6} - 28 q^{7} + 2640 q^{8} - 1536 q^{9} - 4836 q^{10} + 1896 q^{11} - 5046 q^{12} + 10040 q^{13} + 768 q^{14} + 20982 q^{15} - 7868 q^{16} - 56772 q^{17} - 8826 q^{18} + 17084 q^{19} + 71760 q^{20} - 3534 q^{21} + 11220 q^{22} + 30576 q^{23} - 71358 q^{24} - 148106 q^{25} + 61176 q^{26} + 39288 q^{27} - 11932 q^{28} - 73020 q^{29} + 126282 q^{30} + 553460 q^{31} - 384192 q^{32} - 51270 q^{33} - 340788 q^{34} + 49920 q^{35} + 134058 q^{36} - 537208 q^{37} + 103440 q^{38} - 275370 q^{39} + 1029444 q^{40} + 1259436 q^{41} - 20814 q^{42} - 1371700 q^{43} - 174432 q^{44} - 568698 q^{45} + 183300 q^{46} - 1166592 q^{47} + 208146 q^{48} + 1638738 q^{49} - 887700 q^{50} + 1532766 q^{51} - 938188 q^{52} + 856116 q^{53} + 236118 q^{54} + 739284 q^{55} - 168960 q^{56} - 465558 q^{57} - 438276 q^{58} - 2612760 q^{59} - 1937598 q^{60} - 601480 q^{61} + 3321696 q^{62} + 93234 q^{63} - 1318172 q^{64} + 3976440 q^{65} - 307230 q^{66} + 1014332 q^{67} + 5223024 q^{68} - 825630 q^{69} + 299364 q^{70} - 11121264 q^{71} + 1924482 q^{72} - 2738320 q^{73} - 3222312 q^{74} + 3994572 q^{75} - 1586236 q^{76} - 121344 q^{77} - 1651830 q^{78} + 13827284 q^{79} - 3007680 q^{80} - 1062960 q^{81} + 7556460 q^{82} + 8753496 q^{83} + 317874 q^{84} - 22141236 q^{85} - 8229264 q^{86} + 1971462 q^{87} - 2502876 q^{88} + 17056620 q^{89} - 3411798 q^{90} - 652700 q^{91} - 2812992 q^{92} - 14947710 q^{93} - 6999708 q^{94} + 6723600 q^{95} + 10373106 q^{96} + 17653472 q^{97} + 9833364 q^{98} + 1382106 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(471))\)

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
471.8.a \(\chi_{471}(1, \cdot)\) 471.8.a.a 40 1
471.8.a.b 43
471.8.a.c 48
471.8.a.d 51
471.8.b \(\chi_{471}(313, \cdot)\) n/a 186 1
471.8.e \(\chi_{471}(169, \cdot)\) n/a 366 2
471.8.f \(\chi_{471}(185, \cdot)\) n/a 732 2
471.8.h \(\chi_{471}(13, \cdot)\) n/a 370 2
471.8.l \(\chi_{471}(50, \cdot)\) n/a 1468 4
471.8.m \(\chi_{471}(16, \cdot)\) n/a 2208 12
471.8.p \(\chi_{471}(4, \cdot)\) n/a 2232 12
471.8.q \(\chi_{471}(19, \cdot)\) n/a 4392 24
471.8.s \(\chi_{471}(2, \cdot)\) n/a 8784 24
471.8.v \(\chi_{471}(10, \cdot)\) n/a 4440 24
471.8.w \(\chi_{471}(5, \cdot)\) n/a 17616 48

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(471))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(471)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(157))\)\(^{\oplus 2}\)