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Properties

Label 600.8
Level 600
Weight 8
Dimension 26045
Nonzero newspaces 18
Sturm bound 153600
Trace bound 8

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Defining parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(153600\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(600))\).

Total New Old
Modular forms 67872 26209 41663
Cusp forms 66528 26045 40483
Eisenstein series 1344 164 1180

Trace form

\( 26045 q - 14 q^{2} + 13 q^{3} - 260 q^{4} + 58 q^{5} - 34 q^{6} - 2364 q^{7} - 428 q^{8} - 5129 q^{9} + O(q^{10}) \) \( 26045 q - 14 q^{2} + 13 q^{3} - 260 q^{4} + 58 q^{5} - 34 q^{6} - 2364 q^{7} - 428 q^{8} - 5129 q^{9} - 32 q^{10} - 9084 q^{11} + 12340 q^{12} - 63830 q^{13} - 65876 q^{14} + 58712 q^{15} + 196280 q^{16} - 35838 q^{17} - 222318 q^{18} - 258472 q^{19} - 227960 q^{20} - 81520 q^{21} + 373560 q^{22} + 329424 q^{23} + 628896 q^{24} + 24122 q^{25} - 427048 q^{26} - 578591 q^{27} - 1090296 q^{28} + 618162 q^{29} + 989428 q^{30} - 2236172 q^{31} - 1036104 q^{32} + 1191932 q^{33} - 4787308 q^{34} + 1633008 q^{35} - 148264 q^{36} + 919344 q^{37} + 4372648 q^{38} - 7332610 q^{39} + 4367888 q^{40} - 1083350 q^{41} + 351616 q^{42} + 10490904 q^{43} - 8058928 q^{44} + 3793330 q^{45} - 13333824 q^{46} - 1022760 q^{47} + 12601268 q^{48} - 13538527 q^{49} + 7947560 q^{50} - 10635446 q^{51} - 657736 q^{52} + 9581512 q^{53} - 13122886 q^{54} + 9121664 q^{55} + 1643704 q^{56} + 17010512 q^{57} - 2993540 q^{58} - 3990076 q^{59} + 11616388 q^{60} - 15724974 q^{61} + 5767172 q^{62} - 15944552 q^{63} - 18863960 q^{64} + 6541866 q^{65} - 9372380 q^{66} + 45922128 q^{67} + 14608304 q^{68} + 7666488 q^{69} - 19691384 q^{70} - 40089824 q^{71} + 1065224 q^{72} + 61216806 q^{73} - 41765776 q^{74} - 20361048 q^{75} + 10814008 q^{76} - 31805760 q^{77} + 57599588 q^{78} + 62356740 q^{79} + 37919920 q^{80} - 43739529 q^{81} - 85816116 q^{82} + 48410276 q^{83} + 41817812 q^{84} + 129107498 q^{85} - 46716040 q^{86} - 92768718 q^{87} - 210401528 q^{88} + 97169048 q^{89} - 95595412 q^{90} - 71784952 q^{91} - 89423808 q^{92} - 44918736 q^{93} + 45180544 q^{94} + 100893824 q^{95} + 74331628 q^{96} + 126726390 q^{97} + 203689866 q^{98} - 59889316 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.8.a \(\chi_{600}(1, \cdot)\) 600.8.a.a 1 1
600.8.a.b 1
600.8.a.c 1
600.8.a.d 1
600.8.a.e 1
600.8.a.f 2
600.8.a.g 2
600.8.a.h 2
600.8.a.i 2
600.8.a.j 2
600.8.a.k 2
600.8.a.l 3
600.8.a.m 3
600.8.a.n 3
600.8.a.o 3
600.8.a.p 4
600.8.a.q 4
600.8.a.r 4
600.8.a.s 4
600.8.a.t 5
600.8.a.u 5
600.8.a.v 6
600.8.a.w 6
600.8.b \(\chi_{600}(251, \cdot)\) n/a 526 1
600.8.d \(\chi_{600}(349, \cdot)\) n/a 252 1
600.8.f \(\chi_{600}(49, \cdot)\) 600.8.f.a 2 1
600.8.f.b 2
600.8.f.c 2
600.8.f.d 2
600.8.f.e 2
600.8.f.f 4
600.8.f.g 4
600.8.f.h 4
600.8.f.i 4
600.8.f.j 4
600.8.f.k 4
600.8.f.l 6
600.8.f.m 6
600.8.f.n 8
600.8.f.o 8
600.8.h \(\chi_{600}(551, \cdot)\) None 0 1
600.8.k \(\chi_{600}(301, \cdot)\) n/a 266 1
600.8.m \(\chi_{600}(299, \cdot)\) n/a 500 1
600.8.o \(\chi_{600}(599, \cdot)\) None 0 1
600.8.r \(\chi_{600}(257, \cdot)\) n/a 252 2
600.8.s \(\chi_{600}(7, \cdot)\) None 0 2
600.8.v \(\chi_{600}(43, \cdot)\) n/a 504 2
600.8.w \(\chi_{600}(293, \cdot)\) n/a 1000 2
600.8.y \(\chi_{600}(121, \cdot)\) n/a 416 4
600.8.ba \(\chi_{600}(71, \cdot)\) None 0 4
600.8.bc \(\chi_{600}(169, \cdot)\) n/a 424 4
600.8.be \(\chi_{600}(109, \cdot)\) n/a 1680 4
600.8.bg \(\chi_{600}(11, \cdot)\) n/a 3344 4
600.8.bi \(\chi_{600}(119, \cdot)\) None 0 4
600.8.bk \(\chi_{600}(59, \cdot)\) n/a 3344 4
600.8.bm \(\chi_{600}(61, \cdot)\) n/a 1680 4
600.8.bp \(\chi_{600}(53, \cdot)\) n/a 6688 8
600.8.bq \(\chi_{600}(67, \cdot)\) n/a 3360 8
600.8.bt \(\chi_{600}(103, \cdot)\) None 0 8
600.8.bu \(\chi_{600}(17, \cdot)\) n/a 1680 8

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(600)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 2}\)