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Properties

Label 408.4
Level 408
Weight 4
Dimension 5734
Nonzero newspaces 15
Sturm bound 36864
Trace bound 6

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Defining parameters

Level: \( N \) = \( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 15 \)
Sturm bound: \(36864\)
Trace bound: \(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(408))\).

Total New Old
Modular forms 14208 5854 8354
Cusp forms 13440 5734 7706
Eisenstein series 768 120 648

Trace form

\( 5734 q - 4 q^{2} - 18 q^{3} - 72 q^{4} - 28 q^{5} + 12 q^{6} - 40 q^{7} + 152 q^{8} + 62 q^{9} + O(q^{10}) \) \( 5734 q - 4 q^{2} - 18 q^{3} - 72 q^{4} - 28 q^{5} + 12 q^{6} - 40 q^{7} + 152 q^{8} + 62 q^{9} - 104 q^{10} + 56 q^{11} + 96 q^{12} + 148 q^{13} + 200 q^{14} + 20 q^{15} + 160 q^{16} - 134 q^{17} - 364 q^{18} - 160 q^{19} - 112 q^{20} + 144 q^{21} - 928 q^{22} - 672 q^{23} - 1080 q^{24} + 1474 q^{25} - 112 q^{26} + 198 q^{27} - 384 q^{28} - 244 q^{29} + 488 q^{30} - 648 q^{31} + 496 q^{32} - 792 q^{33} + 1300 q^{34} - 1312 q^{35} + 2040 q^{36} - 1404 q^{37} + 1552 q^{38} - 292 q^{39} + 2000 q^{40} - 116 q^{41} - 1528 q^{42} + 1840 q^{43} - 2304 q^{44} + 540 q^{45} - 3568 q^{46} + 336 q^{47} - 5040 q^{48} - 1306 q^{49} - 3940 q^{50} - 1014 q^{51} - 3552 q^{52} - 3172 q^{53} - 4988 q^{54} - 9840 q^{55} - 6672 q^{56} - 1512 q^{57} - 232 q^{58} - 1672 q^{59} + 9856 q^{60} + 2836 q^{61} + 12504 q^{62} + 7688 q^{63} + 13632 q^{64} + 7024 q^{65} + 10800 q^{66} + 10512 q^{67} + 20608 q^{68} + 6192 q^{69} + 11344 q^{70} + 6976 q^{71} + 1832 q^{72} + 484 q^{73} + 1664 q^{74} + 5058 q^{75} - 4880 q^{76} - 3552 q^{77} - 4784 q^{78} - 6120 q^{79} - 15712 q^{80} - 4234 q^{81} - 12168 q^{82} - 13592 q^{83} - 3984 q^{84} - 6884 q^{85} + 1520 q^{86} - 6508 q^{87} + 5120 q^{88} + 932 q^{89} - 9144 q^{90} - 9920 q^{91} - 3456 q^{92} + 1376 q^{93} - 13808 q^{94} - 12784 q^{95} - 12864 q^{96} + 3196 q^{97} - 6708 q^{98} - 896 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(408))\)

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
408.4.a \(\chi_{408}(1, \cdot)\) 408.4.a.a 1 1
408.4.a.b 1
408.4.a.c 2
408.4.a.d 3
408.4.a.e 3
408.4.a.f 3
408.4.a.g 3
408.4.a.h 4
408.4.a.i 4
408.4.c \(\chi_{408}(169, \cdot)\) 408.4.c.a 2 1
408.4.c.b 6
408.4.c.c 6
408.4.c.d 12
408.4.e \(\chi_{408}(239, \cdot)\) None 0 1
408.4.f \(\chi_{408}(205, \cdot)\) 408.4.f.a 48 1
408.4.f.b 48
408.4.h \(\chi_{408}(203, \cdot)\) n/a 212 1
408.4.j \(\chi_{408}(35, \cdot)\) n/a 192 1
408.4.l \(\chi_{408}(373, \cdot)\) n/a 108 1
408.4.o \(\chi_{408}(407, \cdot)\) None 0 1
408.4.q \(\chi_{408}(251, \cdot)\) n/a 424 2
408.4.s \(\chi_{408}(13, \cdot)\) n/a 216 2
408.4.v \(\chi_{408}(217, \cdot)\) 408.4.v.a 24 2
408.4.v.b 28
408.4.x \(\chi_{408}(47, \cdot)\) None 0 2
408.4.ba \(\chi_{408}(25, \cdot)\) n/a 112 4
408.4.bb \(\chi_{408}(263, \cdot)\) None 0 4
408.4.bc \(\chi_{408}(229, \cdot)\) n/a 432 4
408.4.bd \(\chi_{408}(59, \cdot)\) n/a 848 4
408.4.bh \(\chi_{408}(41, \cdot)\) n/a 432 8
408.4.bi \(\chi_{408}(7, \cdot)\) None 0 8
408.4.bl \(\chi_{408}(91, \cdot)\) n/a 864 8
408.4.bm \(\chi_{408}(5, \cdot)\) n/a 1696 8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(408))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(408)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(102))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 2}\)