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Properties

Label 712.6
Level 712
Weight 6
Dimension 44364
Nonzero newspaces 14
Sturm bound 190080
Trace bound 6

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Defining parameters

Level: \( N \) = \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 14 \)
Sturm bound: \(190080\)
Trace bound: \(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(712))\).

Total New Old
Modular forms 79728 44712 35016
Cusp forms 78672 44364 34308
Eisenstein series 1056 348 708

Trace form

\( 44364 q - 84 q^{2} - 128 q^{3} - 128 q^{4} + 148 q^{5} + 144 q^{6} - 232 q^{7} + 408 q^{8} - 162 q^{9} + O(q^{10}) \) \( 44364 q - 84 q^{2} - 128 q^{3} - 128 q^{4} + 148 q^{5} + 144 q^{6} - 232 q^{7} + 408 q^{8} - 162 q^{9} - 1352 q^{10} - 336 q^{11} - 3240 q^{12} - 956 q^{13} + 4680 q^{14} + 3704 q^{15} + 6536 q^{16} + 1820 q^{17} - 9596 q^{18} - 6176 q^{19} - 9336 q^{20} + 960 q^{21} + 11184 q^{22} - 5128 q^{23} + 15496 q^{24} - 7990 q^{25} - 11304 q^{26} + 3352 q^{27} - 10392 q^{28} + 6564 q^{29} + 4168 q^{30} + 37224 q^{31} - 10904 q^{32} - 432 q^{33} + 9456 q^{34} - 3640 q^{35} + 20240 q^{36} - 20652 q^{37} - 32048 q^{38} - 89416 q^{39} - 32152 q^{40} + 26732 q^{41} + 52200 q^{42} + 18288 q^{43} + 58136 q^{44} + 23236 q^{45} - 58488 q^{46} + 62024 q^{47} - 71320 q^{48} + 12630 q^{49} + 94908 q^{50} + 47832 q^{51} + 73032 q^{52} - 23372 q^{53} - 46664 q^{54} - 152680 q^{55} - 81624 q^{56} - 117872 q^{57} - 7656 q^{58} - 33840 q^{59} - 5272 q^{60} + 36964 q^{61} - 69080 q^{62} + 314328 q^{63} + 83752 q^{64} + 109608 q^{65} - 86536 q^{66} + 30976 q^{67} - 20776 q^{68} - 7360 q^{69} + 137768 q^{70} - 349544 q^{71} + 166456 q^{72} - 70356 q^{73} - 35016 q^{74} - 94128 q^{75} - 199976 q^{76} + 5952 q^{77} + 348040 q^{78} + 406536 q^{79} + 70952 q^{80} + 85558 q^{81} - 322352 q^{82} - 134816 q^{83} - 393432 q^{84} - 177304 q^{85} + 36912 q^{86} - 484552 q^{87} + 334344 q^{88} + 12462 q^{89} - 284736 q^{90} + 22856 q^{91} + 98024 q^{92} + 229120 q^{93} + 197480 q^{94} + 969912 q^{95} + 231208 q^{96} + 101244 q^{97} + 233996 q^{98} - 39024 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(712))\)

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
712.6.a \(\chi_{712}(1, \cdot)\) 712.6.a.a 25 1
712.6.a.b 27
712.6.a.c 28
712.6.a.d 30
712.6.b \(\chi_{712}(357, \cdot)\) n/a 440 1
712.6.e \(\chi_{712}(177, \cdot)\) n/a 112 1
712.6.f \(\chi_{712}(533, \cdot)\) n/a 448 1
712.6.j \(\chi_{712}(233, \cdot)\) n/a 226 2
712.6.k \(\chi_{712}(301, \cdot)\) n/a 896 2
712.6.m \(\chi_{712}(571, \cdot)\) n/a 1792 4
712.6.o \(\chi_{712}(215, \cdot)\) None 0 4
712.6.q \(\chi_{712}(97, \cdot)\) n/a 1120 10
712.6.t \(\chi_{712}(85, \cdot)\) n/a 4480 10
712.6.u \(\chi_{712}(25, \cdot)\) n/a 1120 10
712.6.x \(\chi_{712}(45, \cdot)\) n/a 4480 10
712.6.z \(\chi_{712}(5, \cdot)\) n/a 8960 20
712.6.ba \(\chi_{712}(9, \cdot)\) n/a 2260 20
712.6.bd \(\chi_{712}(7, \cdot)\) None 0 40
712.6.bf \(\chi_{712}(3, \cdot)\) n/a 17920 40

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

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(712))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(712)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(89))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(178))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(356))\)\(^{\oplus 2}\)