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Properties

Label 588.8
Level 588
Weight 8
Dimension 27554
Nonzero newspaces 16
Sturm bound 150528
Trace bound 3

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Defining parameters

Level: \( N \) = \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(150528\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(588))\).

Total New Old
Modular forms 66456 27750 38706
Cusp forms 65256 27554 37702
Eisenstein series 1200 196 1004

Trace form

\( 27554 q + 54 q^{3} - 6 q^{4} - 120 q^{5} - 225 q^{6} + 664 q^{7} - 3444 q^{8} - 3438 q^{9} + O(q^{10}) \) \( 27554 q + 54 q^{3} - 6 q^{4} - 120 q^{5} - 225 q^{6} + 664 q^{7} - 3444 q^{8} - 3438 q^{9} + 37614 q^{10} + 33132 q^{11} - 26913 q^{12} - 75228 q^{13} - 39288 q^{14} + 28332 q^{15} + 176562 q^{16} + 65148 q^{17} + 52899 q^{18} - 54576 q^{19} - 155631 q^{21} - 59766 q^{22} + 225672 q^{23} - 185937 q^{24} - 170470 q^{25} + 818940 q^{26} - 78732 q^{27} + 834540 q^{28} - 1262628 q^{29} - 1298589 q^{30} + 493152 q^{31} - 2187420 q^{32} + 2647104 q^{33} - 24138 q^{34} + 317010 q^{35} - 1305771 q^{36} - 4606796 q^{37} + 603252 q^{38} - 3064143 q^{39} - 2846970 q^{40} + 2245032 q^{41} + 1262703 q^{42} + 3137252 q^{43} + 6197136 q^{44} + 1857582 q^{45} + 6297390 q^{46} - 7132704 q^{47} - 8863350 q^{48} - 15585462 q^{49} - 9187500 q^{50} - 430398 q^{51} - 4157418 q^{52} + 7359768 q^{53} + 6664131 q^{54} + 41486250 q^{55} + 2613726 q^{56} + 936576 q^{57} - 8617242 q^{58} - 17181540 q^{59} + 5581083 q^{60} - 30350974 q^{61} + 3586152 q^{63} - 4773306 q^{64} + 3616356 q^{65} - 9236973 q^{66} - 15911720 q^{67} - 7309944 q^{68} + 10587534 q^{69} - 26908446 q^{70} - 19213776 q^{71} - 1745961 q^{72} - 2364276 q^{73} + 26472732 q^{74} + 4881708 q^{75} + 39586158 q^{76} + 36824760 q^{77} + 19676649 q^{78} + 53309680 q^{79} - 112976622 q^{80} + 50995710 q^{81} + 82068624 q^{82} + 32840928 q^{83} + 67854726 q^{84} - 65594340 q^{85} - 24184854 q^{86} - 34232424 q^{87} - 248240088 q^{88} - 44967900 q^{89} - 153733518 q^{90} - 30626252 q^{91} - 80800902 q^{92} - 17379312 q^{93} + 182254176 q^{94} + 119389668 q^{95} + 292308618 q^{96} + 205804668 q^{97} + 461139042 q^{98} + 61750020 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(588))\)

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
588.8.a \(\chi_{588}(1, \cdot)\) 588.8.a.a 1 1
588.8.a.b 1
588.8.a.c 1
588.8.a.d 1
588.8.a.e 2
588.8.a.f 2
588.8.a.g 3
588.8.a.h 3
588.8.a.i 4
588.8.a.j 4
588.8.a.k 5
588.8.a.l 5
588.8.a.m 8
588.8.a.n 8
588.8.b \(\chi_{588}(391, \cdot)\) n/a 280 1
588.8.e \(\chi_{588}(491, \cdot)\) n/a 564 1
588.8.f \(\chi_{588}(293, \cdot)\) 588.8.f.a 2 1
588.8.f.b 36
588.8.f.c 56
588.8.i \(\chi_{588}(361, \cdot)\) 588.8.i.a 2 2
588.8.i.b 2
588.8.i.c 2
588.8.i.d 2
588.8.i.e 2
588.8.i.f 2
588.8.i.g 2
588.8.i.h 2
588.8.i.i 4
588.8.i.j 4
588.8.i.k 4
588.8.i.l 4
588.8.i.m 6
588.8.i.n 6
588.8.i.o 8
588.8.i.p 10
588.8.i.q 16
588.8.i.r 16
588.8.k \(\chi_{588}(509, \cdot)\) n/a 186 2
588.8.n \(\chi_{588}(263, \cdot)\) n/a 1104 2
588.8.o \(\chi_{588}(19, \cdot)\) n/a 560 2
588.8.q \(\chi_{588}(85, \cdot)\) n/a 396 6
588.8.t \(\chi_{588}(41, \cdot)\) n/a 780 6
588.8.u \(\chi_{588}(71, \cdot)\) n/a 4680 6
588.8.x \(\chi_{588}(55, \cdot)\) n/a 2352 6
588.8.y \(\chi_{588}(25, \cdot)\) n/a 780 12
588.8.ba \(\chi_{588}(103, \cdot)\) n/a 4704 12
588.8.bb \(\chi_{588}(11, \cdot)\) n/a 9360 12
588.8.be \(\chi_{588}(5, \cdot)\) n/a 1572 12

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(588))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(588)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(588))\)\(^{\oplus 1}\)