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Properties

Label 600.4
Level 600
Weight 4
Dimension 11115
Nonzero newspaces 18
Sturm bound 76800
Trace bound 8

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Defining parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(76800\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(600))\).

Total New Old
Modular forms 29472 11279 18193
Cusp forms 28128 11115 17013
Eisenstein series 1344 164 1180

Trace form

\( 11115 q + 2 q^{2} - 11 q^{3} - 4 q^{4} - 22 q^{5} - 34 q^{6} + 28 q^{7} - 76 q^{8} - 35 q^{9} + O(q^{10}) \) \( 11115 q + 2 q^{2} - 11 q^{3} - 4 q^{4} - 22 q^{5} - 34 q^{6} + 28 q^{7} - 76 q^{8} - 35 q^{9} - 32 q^{10} + 84 q^{11} + 148 q^{12} + 142 q^{13} + 428 q^{14} - 88 q^{15} - 584 q^{16} - 210 q^{17} - 798 q^{18} - 856 q^{19} - 760 q^{20} - 136 q^{21} - 616 q^{22} + 576 q^{24} + 2 q^{25} + 1528 q^{26} + 217 q^{27} + 1672 q^{28} + 1566 q^{29} - 332 q^{30} + 1516 q^{31} - 1608 q^{32} - 508 q^{33} - 204 q^{34} - 912 q^{35} - 616 q^{36} - 1308 q^{37} + 3368 q^{38} + 14 q^{39} + 3088 q^{40} - 1258 q^{41} + 3232 q^{42} - 3384 q^{43} + 3280 q^{44} - 1910 q^{45} + 1344 q^{46} - 600 q^{47} - 3916 q^{48} + 375 q^{49} - 2840 q^{50} + 2746 q^{51} - 5768 q^{52} - 412 q^{53} + 890 q^{54} - 1376 q^{55} - 1864 q^{56} + 2120 q^{57} - 2500 q^{58} + 1252 q^{59} + 3268 q^{60} - 362 q^{61} - 2108 q^{62} + 3568 q^{63} - 9304 q^{64} + 5586 q^{65} - 3356 q^{66} - 5456 q^{67} - 2960 q^{68} + 2232 q^{69} - 824 q^{70} + 2192 q^{71} - 1048 q^{72} - 398 q^{73} + 10192 q^{74} + 3192 q^{75} + 10360 q^{76} - 2208 q^{77} + 7940 q^{78} + 12188 q^{79} + 6320 q^{80} - 9555 q^{81} - 13716 q^{82} - 6860 q^{83} - 11308 q^{84} - 19302 q^{85} - 8 q^{86} - 6342 q^{87} - 10872 q^{88} + 1228 q^{89} - 1012 q^{90} - 8488 q^{91} + 960 q^{92} - 792 q^{93} + 18688 q^{94} + 5824 q^{95} - 3860 q^{96} + 10770 q^{97} + 23322 q^{98} - 7828 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.4.a \(\chi_{600}(1, \cdot)\) 600.4.a.a 1 1
600.4.a.b 1
600.4.a.c 1
600.4.a.d 1
600.4.a.e 1
600.4.a.f 1
600.4.a.g 1
600.4.a.h 1
600.4.a.i 1
600.4.a.j 1
600.4.a.k 1
600.4.a.l 1
600.4.a.m 1
600.4.a.n 1
600.4.a.o 1
600.4.a.p 1
600.4.a.q 1
600.4.a.r 2
600.4.a.s 2
600.4.a.t 2
600.4.a.u 2
600.4.a.v 2
600.4.a.w 2
600.4.b \(\chi_{600}(251, \cdot)\) n/a 222 1
600.4.d \(\chi_{600}(349, \cdot)\) n/a 108 1
600.4.f \(\chi_{600}(49, \cdot)\) 600.4.f.a 2 1
600.4.f.b 2
600.4.f.c 2
600.4.f.d 2
600.4.f.e 2
600.4.f.f 2
600.4.f.g 2
600.4.f.h 2
600.4.f.i 2
600.4.f.j 4
600.4.f.k 4
600.4.h \(\chi_{600}(551, \cdot)\) None 0 1
600.4.k \(\chi_{600}(301, \cdot)\) n/a 114 1
600.4.m \(\chi_{600}(299, \cdot)\) n/a 212 1
600.4.o \(\chi_{600}(599, \cdot)\) None 0 1
600.4.r \(\chi_{600}(257, \cdot)\) n/a 108 2
600.4.s \(\chi_{600}(7, \cdot)\) None 0 2
600.4.v \(\chi_{600}(43, \cdot)\) n/a 216 2
600.4.w \(\chi_{600}(293, \cdot)\) n/a 424 2
600.4.y \(\chi_{600}(121, \cdot)\) n/a 176 4
600.4.ba \(\chi_{600}(71, \cdot)\) None 0 4
600.4.bc \(\chi_{600}(169, \cdot)\) n/a 184 4
600.4.be \(\chi_{600}(109, \cdot)\) n/a 720 4
600.4.bg \(\chi_{600}(11, \cdot)\) n/a 1424 4
600.4.bi \(\chi_{600}(119, \cdot)\) None 0 4
600.4.bk \(\chi_{600}(59, \cdot)\) n/a 1424 4
600.4.bm \(\chi_{600}(61, \cdot)\) n/a 720 4
600.4.bp \(\chi_{600}(53, \cdot)\) n/a 2848 8
600.4.bq \(\chi_{600}(67, \cdot)\) n/a 1440 8
600.4.bt \(\chi_{600}(103, \cdot)\) None 0 8
600.4.bu \(\chi_{600}(17, \cdot)\) n/a 720 8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

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