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Properties

Label 448.8
Level 448
Weight 8
Dimension 23074
Nonzero newspaces 16
Sturm bound 98304
Trace bound 25

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Defining parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(98304\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(448))\).

Total New Old
Modular forms 43440 23294 20146
Cusp forms 42576 23074 19502
Eisenstein series 864 220 644

Trace form

\( 23074 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 4334 q^{9} + O(q^{10}) \) \( 23074 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 4334 q^{9} - 32 q^{10} + 2384 q^{11} - 32 q^{12} + 14096 q^{13} - 40 q^{14} - 54064 q^{15} - 32 q^{16} + 11576 q^{17} - 32 q^{18} + 121144 q^{19} - 32 q^{20} - 36220 q^{21} - 548992 q^{22} - 20 q^{23} + 1196048 q^{24} + 27218 q^{25} - 1455952 q^{26} - 467364 q^{27} - 390960 q^{28} + 206672 q^{29} + 3479808 q^{30} + 714940 q^{31} + 2142608 q^{32} + 800852 q^{33} - 806352 q^{34} - 252032 q^{35} - 8139120 q^{36} - 2930384 q^{37} - 2489872 q^{38} - 20 q^{39} + 5372688 q^{40} + 5935064 q^{41} + 3858880 q^{42} + 1139452 q^{43} - 6740976 q^{44} - 2835960 q^{45} - 32 q^{46} - 4152948 q^{47} - 32 q^{48} - 1022166 q^{49} + 4634752 q^{50} + 17192308 q^{51} - 20409920 q^{52} + 7731952 q^{53} + 2519392 q^{54} - 16764064 q^{55} + 10737232 q^{56} - 8941112 q^{57} + 20716240 q^{58} - 17074048 q^{59} - 8672864 q^{60} - 614896 q^{61} - 20697088 q^{62} + 28298848 q^{63} - 45674384 q^{64} + 1832940 q^{65} - 15030496 q^{66} - 18891656 q^{67} + 17500096 q^{68} + 3730840 q^{69} + 35990264 q^{70} - 49394872 q^{71} + 45454576 q^{72} + 5672920 q^{73} - 8428816 q^{74} + 57467160 q^{75} - 77885856 q^{76} - 10982204 q^{77} - 137665856 q^{78} - 18150356 q^{79} + 156724880 q^{80} + 20128894 q^{81} + 98629248 q^{82} + 9062376 q^{83} - 17056856 q^{84} - 8904192 q^{85} - 139071120 q^{86} - 20 q^{87} - 105035712 q^{88} - 41791432 q^{89} - 63468032 q^{90} - 7294892 q^{91} + 108881472 q^{92} - 27099584 q^{93} + 149383072 q^{94} + 87208572 q^{95} + 225737936 q^{96} + 157333784 q^{97} + 62768992 q^{98} + 11263484 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(448))\)

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
448.8.a \(\chi_{448}(1, \cdot)\) 448.8.a.a 1 1
448.8.a.b 1
448.8.a.c 1
448.8.a.d 1
448.8.a.e 1
448.8.a.f 1
448.8.a.g 1
448.8.a.h 1
448.8.a.i 1
448.8.a.j 1
448.8.a.k 2
448.8.a.l 2
448.8.a.m 2
448.8.a.n 2
448.8.a.o 2
448.8.a.p 2
448.8.a.q 2
448.8.a.r 2
448.8.a.s 2
448.8.a.t 2
448.8.a.u 3
448.8.a.v 3
448.8.a.w 3
448.8.a.x 3
448.8.a.y 4
448.8.a.z 4
448.8.a.ba 5
448.8.a.bb 5
448.8.a.bc 6
448.8.a.bd 6
448.8.a.be 6
448.8.a.bf 6
448.8.b \(\chi_{448}(225, \cdot)\) 448.8.b.a 14 1
448.8.b.b 14
448.8.b.c 28
448.8.b.d 28
448.8.e \(\chi_{448}(223, \cdot)\) n/a 112 1
448.8.f \(\chi_{448}(447, \cdot)\) n/a 110 1
448.8.i \(\chi_{448}(65, \cdot)\) n/a 220 2
448.8.j \(\chi_{448}(111, \cdot)\) n/a 220 2
448.8.m \(\chi_{448}(113, \cdot)\) n/a 168 2
448.8.p \(\chi_{448}(255, \cdot)\) n/a 220 2
448.8.q \(\chi_{448}(31, \cdot)\) n/a 224 2
448.8.t \(\chi_{448}(289, \cdot)\) n/a 224 2
448.8.u \(\chi_{448}(57, \cdot)\) None 0 4
448.8.x \(\chi_{448}(55, \cdot)\) None 0 4
448.8.z \(\chi_{448}(47, \cdot)\) n/a 440 4
448.8.ba \(\chi_{448}(81, \cdot)\) n/a 440 4
448.8.bc \(\chi_{448}(29, \cdot)\) n/a 2688 8
448.8.bd \(\chi_{448}(27, \cdot)\) n/a 3568 8
448.8.bh \(\chi_{448}(9, \cdot)\) None 0 8
448.8.bi \(\chi_{448}(87, \cdot)\) None 0 8
448.8.bm \(\chi_{448}(3, \cdot)\) n/a 7136 16
448.8.bn \(\chi_{448}(37, \cdot)\) n/a 7136 16

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(448))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(448)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 14}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(448))\)\(^{\oplus 1}\)