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Properties

Label 399.4
Level 399
Weight 4
Dimension 12180
Nonzero newspaces 32
Sturm bound 46080
Trace bound 12

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Defining parameters

Level: \( N \) = \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 32 \)
Sturm bound: \(46080\)
Trace bound: \(12\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(399))\).

Total New Old
Modular forms 17712 12516 5196
Cusp forms 16848 12180 4668
Eisenstein series 864 336 528

Trace form

\( 12180 q - 42 q^{3} - 60 q^{4} + 48 q^{5} + 36 q^{6} + 18 q^{7} - 108 q^{8} - 114 q^{9} + O(q^{10}) \) \( 12180 q - 42 q^{3} - 60 q^{4} + 48 q^{5} + 36 q^{6} + 18 q^{7} - 108 q^{8} - 114 q^{9} - 168 q^{10} - 420 q^{12} - 480 q^{13} + 552 q^{14} + 522 q^{15} + 1908 q^{16} + 600 q^{17} - 594 q^{18} + 882 q^{19} + 216 q^{20} - 963 q^{21} - 396 q^{22} + 264 q^{23} - 1188 q^{24} - 948 q^{25} - 2148 q^{26} - 1458 q^{27} - 2460 q^{28} - 1224 q^{29} + 630 q^{30} + 1068 q^{31} + 3696 q^{32} + 3402 q^{33} + 4980 q^{34} + 2064 q^{35} + 5838 q^{36} - 288 q^{37} + 7614 q^{38} + 1284 q^{39} + 4620 q^{40} + 1416 q^{41} - 3159 q^{42} - 1476 q^{43} - 5268 q^{44} - 2286 q^{45} - 7788 q^{46} - 5400 q^{47} - 5106 q^{48} + 1182 q^{49} - 6336 q^{50} + 1152 q^{51} + 9288 q^{52} + 4200 q^{53} + 2880 q^{54} + 1968 q^{55} - 1380 q^{56} - 4980 q^{57} - 7200 q^{58} + 672 q^{59} - 22086 q^{60} - 22548 q^{61} - 22884 q^{62} - 6087 q^{63} - 25152 q^{64} - 8568 q^{65} + 936 q^{66} - 1044 q^{67} + 7860 q^{68} + 13428 q^{69} + 16638 q^{70} + 8904 q^{71} + 23652 q^{72} + 30936 q^{73} + 26484 q^{74} + 17154 q^{75} + 32472 q^{76} + 9684 q^{77} - 1638 q^{78} + 11604 q^{79} - 15408 q^{80} - 14346 q^{81} - 6156 q^{82} - 9768 q^{83} - 17361 q^{84} - 13260 q^{85} - 15984 q^{86} - 6084 q^{87} - 1056 q^{88} - 12696 q^{89} + 12834 q^{90} - 11634 q^{91} - 708 q^{92} + 618 q^{93} + 6732 q^{94} + 8808 q^{95} + 59616 q^{96} + 8832 q^{97} + 23772 q^{98} + 31680 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(399))\)

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
399.4.a \(\chi_{399}(1, \cdot)\) 399.4.a.a 1 1
399.4.a.b 1
399.4.a.c 2
399.4.a.d 4
399.4.a.e 5
399.4.a.f 6
399.4.a.g 7
399.4.a.h 8
399.4.a.i 9
399.4.a.j 9
399.4.c \(\chi_{399}(265, \cdot)\) 399.4.c.a 40 1
399.4.c.b 40
399.4.d \(\chi_{399}(20, \cdot)\) n/a 144 1
399.4.f \(\chi_{399}(113, \cdot)\) n/a 120 1
399.4.i \(\chi_{399}(163, \cdot)\) n/a 160 2
399.4.j \(\chi_{399}(58, \cdot)\) n/a 144 2
399.4.k \(\chi_{399}(64, \cdot)\) n/a 120 2
399.4.l \(\chi_{399}(121, \cdot)\) n/a 160 2
399.4.m \(\chi_{399}(145, \cdot)\) n/a 160 2
399.4.p \(\chi_{399}(26, \cdot)\) n/a 312 2
399.4.t \(\chi_{399}(8, \cdot)\) n/a 240 2
399.4.w \(\chi_{399}(170, \cdot)\) n/a 312 2
399.4.x \(\chi_{399}(65, \cdot)\) n/a 312 2
399.4.z \(\chi_{399}(83, \cdot)\) n/a 312 2
399.4.bc \(\chi_{399}(248, \cdot)\) n/a 288 2
399.4.bd \(\chi_{399}(68, \cdot)\) n/a 312 2
399.4.bg \(\chi_{399}(31, \cdot)\) n/a 160 2
399.4.bh \(\chi_{399}(94, \cdot)\) n/a 160 2
399.4.bk \(\chi_{399}(160, \cdot)\) n/a 160 2
399.4.bm \(\chi_{399}(107, \cdot)\) n/a 312 2
399.4.bo \(\chi_{399}(43, \cdot)\) n/a 360 6
399.4.bp \(\chi_{399}(130, \cdot)\) n/a 480 6
399.4.bq \(\chi_{399}(4, \cdot)\) n/a 480 6
399.4.br \(\chi_{399}(2, \cdot)\) n/a 936 6
399.4.bv \(\chi_{399}(86, \cdot)\) n/a 936 6
399.4.bw \(\chi_{399}(29, \cdot)\) n/a 720 6
399.4.cb \(\chi_{399}(5, \cdot)\) n/a 936 6
399.4.cc \(\chi_{399}(13, \cdot)\) n/a 480 6
399.4.cd \(\chi_{399}(52, \cdot)\) n/a 480 6
399.4.ci \(\chi_{399}(17, \cdot)\) n/a 936 6
399.4.cj \(\chi_{399}(62, \cdot)\) n/a 936 6
399.4.ck \(\chi_{399}(10, \cdot)\) n/a 480 6

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(399))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(399)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(133))\)\(^{\oplus 2}\)