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Properties

Label 440.4
Level 440
Weight 4
Dimension 8656
Nonzero newspaces 18
Sturm bound 46080
Trace bound 6

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Defining parameters

Level: \( N \) = \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(46080\)
Trace bound: \(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(440))\).

Total New Old
Modular forms 17760 8872 8888
Cusp forms 16800 8656 8144
Eisenstein series 960 216 744

Trace form

\( 8656 q - 20 q^{2} - 28 q^{3} - 60 q^{4} + 18 q^{5} + 68 q^{6} - 4 q^{7} + 148 q^{8} - 166 q^{9} + O(q^{10}) \) \( 8656 q - 20 q^{2} - 28 q^{3} - 60 q^{4} + 18 q^{5} + 68 q^{6} - 4 q^{7} + 148 q^{8} - 166 q^{9} - 134 q^{10} - 168 q^{11} - 472 q^{12} - 20 q^{13} - 212 q^{14} + 572 q^{15} + 244 q^{16} + 216 q^{17} + 948 q^{18} + 162 q^{19} + 962 q^{20} - 472 q^{21} + 76 q^{22} - 1476 q^{23} - 516 q^{24} - 374 q^{25} + 340 q^{26} - 790 q^{27} - 388 q^{28} - 56 q^{29} + 116 q^{30} + 1000 q^{31} - 2520 q^{32} - 1594 q^{33} - 4256 q^{34} - 52 q^{35} - 3804 q^{36} + 988 q^{37} - 1632 q^{38} + 4612 q^{39} - 524 q^{40} + 2952 q^{41} + 5192 q^{42} + 4372 q^{43} + 5552 q^{44} + 616 q^{45} + 4576 q^{46} + 2816 q^{47} + 5760 q^{48} - 3874 q^{49} - 144 q^{50} - 6810 q^{51} - 1168 q^{52} + 1556 q^{53} - 3076 q^{54} - 1196 q^{55} - 5160 q^{56} + 958 q^{57} - 1240 q^{58} + 1258 q^{59} + 4976 q^{60} + 2132 q^{61} + 4556 q^{62} + 912 q^{63} + 9900 q^{64} - 2776 q^{65} + 3860 q^{66} - 2496 q^{67} + 2276 q^{68} - 7040 q^{69} + 1016 q^{70} - 12932 q^{71} + 4392 q^{72} - 9264 q^{73} + 1872 q^{74} - 12620 q^{75} + 4 q^{76} + 3660 q^{77} - 10440 q^{78} - 3636 q^{79} - 13440 q^{80} + 9964 q^{81} - 23252 q^{82} + 7158 q^{83} - 34724 q^{84} + 2896 q^{85} - 33436 q^{86} + 17488 q^{87} - 29076 q^{88} + 4736 q^{89} - 8150 q^{90} + 19776 q^{91} + 6060 q^{92} + 100 q^{93} + 2668 q^{94} + 5050 q^{95} + 22288 q^{96} + 170 q^{97} + 27200 q^{98} - 4548 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(440))\)

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
440.4.a \(\chi_{440}(1, \cdot)\) 440.4.a.a 1 1
440.4.a.b 1
440.4.a.c 1
440.4.a.d 1
440.4.a.e 2
440.4.a.f 3
440.4.a.g 3
440.4.a.h 4
440.4.a.i 4
440.4.a.j 5
440.4.a.k 5
440.4.b \(\chi_{440}(89, \cdot)\) 440.4.b.a 2 1
440.4.b.b 20
440.4.b.c 24
440.4.c \(\chi_{440}(219, \cdot)\) n/a 212 1
440.4.f \(\chi_{440}(351, \cdot)\) None 0 1
440.4.g \(\chi_{440}(221, \cdot)\) n/a 120 1
440.4.l \(\chi_{440}(309, \cdot)\) n/a 180 1
440.4.m \(\chi_{440}(439, \cdot)\) None 0 1
440.4.p \(\chi_{440}(131, \cdot)\) n/a 144 1
440.4.r \(\chi_{440}(67, \cdot)\) n/a 360 2
440.4.t \(\chi_{440}(197, \cdot)\) n/a 424 2
440.4.v \(\chi_{440}(153, \cdot)\) n/a 108 2
440.4.x \(\chi_{440}(23, \cdot)\) None 0 2
440.4.y \(\chi_{440}(81, \cdot)\) n/a 144 4
440.4.z \(\chi_{440}(51, \cdot)\) n/a 576 4
440.4.bc \(\chi_{440}(39, \cdot)\) None 0 4
440.4.bd \(\chi_{440}(69, \cdot)\) n/a 848 4
440.4.bi \(\chi_{440}(141, \cdot)\) n/a 576 4
440.4.bj \(\chi_{440}(151, \cdot)\) None 0 4
440.4.bm \(\chi_{440}(19, \cdot)\) n/a 848 4
440.4.bn \(\chi_{440}(9, \cdot)\) n/a 216 4
440.4.bo \(\chi_{440}(17, \cdot)\) n/a 432 8
440.4.bq \(\chi_{440}(47, \cdot)\) None 0 8
440.4.bs \(\chi_{440}(3, \cdot)\) n/a 1696 8
440.4.bu \(\chi_{440}(13, \cdot)\) n/a 1696 8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(440))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(440)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(110))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(440))\)\(^{\oplus 1}\)