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Properties

Label 400.10
Level 400
Weight 10
Dimension 22304
Nonzero newspaces 14
Sturm bound 96000
Trace bound 3

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Defining parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 14 \)
Sturm bound: \(96000\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(400))\).

Total New Old
Modular forms 43592 22489 21103
Cusp forms 42808 22304 20504
Eisenstein series 784 185 599

Trace form

\( 22304 q - 26 q^{2} - 100 q^{3} + 144 q^{4} - 40 q^{5} - 2232 q^{6} + 1358 q^{7} - 740 q^{8} + 5810 q^{9} + O(q^{10}) \) \( 22304 q - 26 q^{2} - 100 q^{3} + 144 q^{4} - 40 q^{5} - 2232 q^{6} + 1358 q^{7} - 740 q^{8} + 5810 q^{9} - 32 q^{10} + 54836 q^{11} + 217132 q^{12} - 387744 q^{13} - 133564 q^{14} + 120948 q^{15} - 817296 q^{16} - 643938 q^{17} + 2179878 q^{18} + 2350456 q^{19} - 32 q^{20} - 2834126 q^{21} - 34324 q^{22} - 4156890 q^{23} + 1863832 q^{24} + 3443520 q^{25} - 6347848 q^{26} + 226514 q^{27} - 564800 q^{28} + 5344220 q^{29} + 33356416 q^{30} - 10020030 q^{31} - 104299296 q^{32} - 23984174 q^{33} + 120948764 q^{34} + 19120284 q^{35} + 167171268 q^{36} - 12628496 q^{37} - 189471208 q^{38} - 27600550 q^{39} - 155497712 q^{40} - 24274082 q^{41} + 364715776 q^{42} - 5854680 q^{43} + 333591868 q^{44} + 98122170 q^{45} - 96232156 q^{46} - 114174914 q^{47} - 882997424 q^{48} - 291904018 q^{49} + 65998728 q^{50} + 183558344 q^{51} + 810723484 q^{52} - 681177200 q^{53} - 55932152 q^{54} - 180661450 q^{55} + 26511216 q^{56} + 275759366 q^{57} - 178271808 q^{58} + 742756028 q^{59} + 2097120 q^{60} + 80040404 q^{61} - 136788968 q^{62} - 1441821950 q^{63} - 481588584 q^{64} - 445934924 q^{65} - 1724244724 q^{66} - 1523198644 q^{67} + 2795839720 q^{68} + 503835034 q^{69} + 795897400 q^{70} + 2299499474 q^{71} - 4601012100 q^{72} - 1380516214 q^{73} - 4748657788 q^{74} + 1906227688 q^{75} - 390432828 q^{76} - 550003054 q^{77} + 8102483284 q^{78} - 6217762318 q^{79} + 2874835248 q^{80} - 1634965292 q^{81} - 138802840 q^{82} + 6582184112 q^{83} - 11192944768 q^{84} + 703084976 q^{85} - 6880445732 q^{86} + 4950021794 q^{87} - 721213680 q^{88} - 343553538 q^{89} + 7281234088 q^{90} - 8866156562 q^{91} + 9684039152 q^{92} + 4599700718 q^{93} - 3529189944 q^{94} + 8476818050 q^{95} - 25745861880 q^{96} - 4918300762 q^{97} + 4829660450 q^{98} - 13149876674 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(400))\)

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
400.10.a \(\chi_{400}(1, \cdot)\) 400.10.a.a 1 1
400.10.a.b 1
400.10.a.c 1
400.10.a.d 1
400.10.a.e 1
400.10.a.f 1
400.10.a.g 1
400.10.a.h 1
400.10.a.i 1
400.10.a.j 1
400.10.a.k 1
400.10.a.l 2
400.10.a.m 2
400.10.a.n 2
400.10.a.o 2
400.10.a.p 2
400.10.a.q 2
400.10.a.r 2
400.10.a.s 2
400.10.a.t 2
400.10.a.u 3
400.10.a.v 3
400.10.a.w 3
400.10.a.x 3
400.10.a.y 3
400.10.a.z 4
400.10.a.ba 4
400.10.a.bb 4
400.10.a.bc 4
400.10.a.bd 5
400.10.a.be 5
400.10.a.bf 7
400.10.a.bg 7
400.10.c \(\chi_{400}(49, \cdot)\) 400.10.c.a 2 1
400.10.c.b 2
400.10.c.c 2
400.10.c.d 2
400.10.c.e 2
400.10.c.f 2
400.10.c.g 2
400.10.c.h 2
400.10.c.i 2
400.10.c.j 2
400.10.c.k 4
400.10.c.l 4
400.10.c.m 4
400.10.c.n 4
400.10.c.o 4
400.10.c.p 4
400.10.c.q 6
400.10.c.r 6
400.10.c.s 6
400.10.c.t 8
400.10.c.u 10
400.10.d \(\chi_{400}(201, \cdot)\) None 0 1
400.10.f \(\chi_{400}(249, \cdot)\) None 0 1
400.10.j \(\chi_{400}(43, \cdot)\) n/a 644 2
400.10.l \(\chi_{400}(101, \cdot)\) n/a 678 2
400.10.n \(\chi_{400}(143, \cdot)\) n/a 162 2
400.10.o \(\chi_{400}(7, \cdot)\) None 0 2
400.10.q \(\chi_{400}(149, \cdot)\) n/a 644 2
400.10.s \(\chi_{400}(107, \cdot)\) n/a 644 2
400.10.u \(\chi_{400}(81, \cdot)\) n/a 536 4
400.10.w \(\chi_{400}(9, \cdot)\) None 0 4
400.10.y \(\chi_{400}(129, \cdot)\) n/a 536 4
400.10.bb \(\chi_{400}(41, \cdot)\) None 0 4
400.10.bd \(\chi_{400}(3, \cdot)\) n/a 4304 8
400.10.be \(\chi_{400}(21, \cdot)\) n/a 4304 8
400.10.bh \(\chi_{400}(23, \cdot)\) None 0 8
400.10.bi \(\chi_{400}(47, \cdot)\) n/a 1080 8
400.10.bl \(\chi_{400}(29, \cdot)\) n/a 4304 8
400.10.bm \(\chi_{400}(67, \cdot)\) n/a 4304 8

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

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(400)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)