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Properties

Label 448.4
Level 448
Weight 4
Dimension 9826
Nonzero newspaces 16
Sturm bound 49152
Trace bound 25

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Defining parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(49152\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(448))\).

Total New Old
Modular forms 18864 10046 8818
Cusp forms 18000 9826 8174
Eisenstein series 864 220 644

Trace form

\( 9826 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 14 q^{9} + O(q^{10}) \) \( 9826 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 14 q^{9} - 32 q^{10} + 16 q^{11} - 32 q^{12} - 176 q^{13} - 40 q^{14} - 304 q^{15} - 32 q^{16} - 264 q^{17} - 32 q^{18} - 72 q^{19} - 32 q^{20} - 28 q^{21} - 1024 q^{22} - 20 q^{23} - 2032 q^{24} - 142 q^{25} + 48 q^{26} + 348 q^{27} + 720 q^{28} + 720 q^{29} + 4608 q^{30} + 700 q^{31} + 2448 q^{32} + 1940 q^{33} + 1968 q^{34} + 448 q^{35} + 1680 q^{36} + 1008 q^{37} - 912 q^{38} - 20 q^{39} - 3312 q^{40} - 2088 q^{41} - 3200 q^{42} - 1732 q^{43} - 2032 q^{44} - 3000 q^{45} - 32 q^{46} - 1908 q^{47} - 32 q^{48} - 790 q^{49} + 5632 q^{50} - 8972 q^{51} + 6592 q^{52} - 848 q^{53} + 3424 q^{54} - 1184 q^{55} - 432 q^{56} + 2056 q^{57} - 4784 q^{58} + 8896 q^{59} - 9824 q^{60} + 2128 q^{61} - 6016 q^{62} + 7648 q^{63} - 12176 q^{64} + 4460 q^{65} - 11104 q^{66} + 12024 q^{67} - 4160 q^{68} + 1240 q^{69} - 2056 q^{70} + 840 q^{71} + 1264 q^{72} - 2088 q^{73} + 5232 q^{74} - 14760 q^{75} + 11872 q^{76} - 3164 q^{77} + 3904 q^{78} - 20180 q^{79} - 8560 q^{80} - 14018 q^{81} - 13952 q^{82} - 5144 q^{83} - 4184 q^{84} - 3392 q^{85} + 1008 q^{86} - 20 q^{87} + 6208 q^{88} + 7096 q^{89} + 18688 q^{90} - 812 q^{91} + 25152 q^{92} + 5632 q^{93} + 17824 q^{94} + 2172 q^{95} + 25808 q^{96} + 9176 q^{97} + 12064 q^{98} + 4028 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(448))\)

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
448.4.a \(\chi_{448}(1, \cdot)\) 448.4.a.a 1 1
448.4.a.b 1
448.4.a.c 1
448.4.a.d 1
448.4.a.e 1
448.4.a.f 1
448.4.a.g 1
448.4.a.h 1
448.4.a.i 1
448.4.a.j 1
448.4.a.k 1
448.4.a.l 1
448.4.a.m 1
448.4.a.n 1
448.4.a.o 1
448.4.a.p 1
448.4.a.q 2
448.4.a.r 2
448.4.a.s 2
448.4.a.t 2
448.4.a.u 3
448.4.a.v 3
448.4.a.w 3
448.4.a.x 3
448.4.b \(\chi_{448}(225, \cdot)\) 448.4.b.a 4 1
448.4.b.b 4
448.4.b.c 6
448.4.b.d 6
448.4.b.e 8
448.4.b.f 8
448.4.e \(\chi_{448}(223, \cdot)\) 448.4.e.a 8 1
448.4.e.b 8
448.4.e.c 8
448.4.e.d 24
448.4.f \(\chi_{448}(447, \cdot)\) 448.4.f.a 2 1
448.4.f.b 4
448.4.f.c 8
448.4.f.d 8
448.4.f.e 24
448.4.i \(\chi_{448}(65, \cdot)\) 448.4.i.a 2 2
448.4.i.b 2
448.4.i.c 2
448.4.i.d 2
448.4.i.e 2
448.4.i.f 2
448.4.i.g 4
448.4.i.h 4
448.4.i.i 4
448.4.i.j 6
448.4.i.k 6
448.4.i.l 6
448.4.i.m 6
448.4.i.n 8
448.4.i.o 12
448.4.i.p 12
448.4.i.q 12
448.4.j \(\chi_{448}(111, \cdot)\) 448.4.j.a 4 2
448.4.j.b 88
448.4.m \(\chi_{448}(113, \cdot)\) 448.4.m.a 34 2
448.4.m.b 38
448.4.p \(\chi_{448}(255, \cdot)\) 448.4.p.a 2 2
448.4.p.b 2
448.4.p.c 2
448.4.p.d 2
448.4.p.e 4
448.4.p.f 6
448.4.p.g 6
448.4.p.h 20
448.4.p.i 48
448.4.q \(\chi_{448}(31, \cdot)\) 448.4.q.a 32 2
448.4.q.b 32
448.4.q.c 32
448.4.t \(\chi_{448}(289, \cdot)\) 448.4.t.a 32 2
448.4.t.b 32
448.4.t.c 32
448.4.u \(\chi_{448}(57, \cdot)\) None 0 4
448.4.x \(\chi_{448}(55, \cdot)\) None 0 4
448.4.z \(\chi_{448}(47, \cdot)\) n/a 184 4
448.4.ba \(\chi_{448}(81, \cdot)\) n/a 184 4
448.4.bc \(\chi_{448}(29, \cdot)\) n/a 1152 8
448.4.bd \(\chi_{448}(27, \cdot)\) n/a 1520 8
448.4.bh \(\chi_{448}(9, \cdot)\) None 0 8
448.4.bi \(\chi_{448}(87, \cdot)\) None 0 8
448.4.bm \(\chi_{448}(3, \cdot)\) n/a 3040 16
448.4.bn \(\chi_{448}(37, \cdot)\) n/a 3040 16

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(448))\) into lower level spaces

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