Properties

Label 208.8
Level 208
Weight 8
Dimension 5180
Nonzero newspaces 14
Sturm bound 21504
Trace bound 5

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Defining parameters

Level: \( N \) = \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 14 \)
Sturm bound: \(21504\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(208))\).

Total New Old
Modular forms 9576 5278 4298
Cusp forms 9240 5180 4060
Eisenstein series 336 98 238

Trace form

\( 5180 q - 20 q^{2} - 70 q^{3} + 344 q^{4} - 302 q^{5} + 328 q^{6} - 1346 q^{7} + 1984 q^{8} - 5156 q^{9} + O(q^{10}) \) \( 5180 q - 20 q^{2} - 70 q^{3} + 344 q^{4} - 302 q^{5} + 328 q^{6} - 1346 q^{7} + 1984 q^{8} - 5156 q^{9} - 25968 q^{10} - 11614 q^{11} + 54688 q^{12} + 3250 q^{13} - 44584 q^{14} + 100566 q^{15} - 26696 q^{16} + 25382 q^{17} + 58668 q^{18} - 203766 q^{19} - 92432 q^{20} - 2054 q^{21} + 88672 q^{22} + 303966 q^{23} + 118408 q^{24} - 2710 q^{25} + 35936 q^{26} - 299332 q^{27} - 440648 q^{28} - 136782 q^{29} - 55296 q^{30} + 161870 q^{31} + 497240 q^{32} - 171854 q^{33} - 1849344 q^{34} - 900586 q^{35} + 2030992 q^{36} - 208670 q^{37} + 10920 q^{38} - 1176358 q^{39} + 2243904 q^{40} + 2982006 q^{41} - 249224 q^{42} - 2936926 q^{43} + 1517216 q^{44} - 6227322 q^{45} - 4671344 q^{46} + 2424010 q^{47} - 8282600 q^{48} + 5824936 q^{49} + 1574364 q^{50} + 5126228 q^{51} - 410348 q^{52} + 1064564 q^{53} + 14809000 q^{54} - 6921674 q^{55} + 8186296 q^{56} - 7173302 q^{57} + 3721400 q^{58} + 4619070 q^{59} - 10294680 q^{60} - 5558066 q^{61} - 12548216 q^{62} - 6966714 q^{63} - 20904472 q^{64} + 4483058 q^{65} - 21875672 q^{66} + 9124046 q^{67} + 14807496 q^{68} + 21510618 q^{69} + 44281576 q^{70} - 13347522 q^{71} + 50587872 q^{72} + 12741790 q^{73} + 26354064 q^{74} + 8162940 q^{75} - 35628544 q^{76} - 27474956 q^{77} - 37993940 q^{78} + 9891964 q^{79} - 80094568 q^{80} - 5311506 q^{81} - 45716504 q^{82} - 11615270 q^{83} + 80084416 q^{84} + 71473370 q^{85} + 147825024 q^{86} - 35902686 q^{87} - 14940880 q^{88} - 78086754 q^{89} - 101836608 q^{90} - 6046006 q^{91} + 14392368 q^{92} + 135754946 q^{93} + 20612488 q^{94} + 76676466 q^{95} + 107371832 q^{96} + 24529574 q^{97} + 1449372 q^{98} - 79676918 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(208))\)

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
208.8.a \(\chi_{208}(1, \cdot)\) 208.8.a.a 1 1
208.8.a.b 1
208.8.a.c 1
208.8.a.d 1
208.8.a.e 1
208.8.a.f 2
208.8.a.g 2
208.8.a.h 2
208.8.a.i 3
208.8.a.j 3
208.8.a.k 4
208.8.a.l 4
208.8.a.m 5
208.8.a.n 6
208.8.a.o 6
208.8.b \(\chi_{208}(105, \cdot)\) None 0 1
208.8.e \(\chi_{208}(25, \cdot)\) None 0 1
208.8.f \(\chi_{208}(129, \cdot)\) 208.8.f.a 6 1
208.8.f.b 8
208.8.f.c 10
208.8.f.d 24
208.8.i \(\chi_{208}(81, \cdot)\) 208.8.i.a 6 2
208.8.i.b 8
208.8.i.c 16
208.8.i.d 16
208.8.i.e 24
208.8.i.f 26
208.8.k \(\chi_{208}(31, \cdot)\) 208.8.k.a 2 2
208.8.k.b 28
208.8.k.c 68
208.8.l \(\chi_{208}(83, \cdot)\) n/a 388 2
208.8.n \(\chi_{208}(53, \cdot)\) n/a 336 2
208.8.p \(\chi_{208}(77, \cdot)\) n/a 388 2
208.8.s \(\chi_{208}(99, \cdot)\) n/a 388 2
208.8.u \(\chi_{208}(135, \cdot)\) None 0 2
208.8.w \(\chi_{208}(17, \cdot)\) 208.8.w.a 14 2
208.8.w.b 16
208.8.w.c 18
208.8.w.d 48
208.8.z \(\chi_{208}(9, \cdot)\) None 0 2
208.8.ba \(\chi_{208}(121, \cdot)\) None 0 2
208.8.bc \(\chi_{208}(7, \cdot)\) None 0 4
208.8.bf \(\chi_{208}(11, \cdot)\) n/a 776 4
208.8.bh \(\chi_{208}(69, \cdot)\) n/a 776 4
208.8.bj \(\chi_{208}(29, \cdot)\) n/a 776 4
208.8.bk \(\chi_{208}(115, \cdot)\) n/a 776 4
208.8.bm \(\chi_{208}(15, \cdot)\) n/a 196 4

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(208))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(208)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(208))\)\(^{\oplus 1}\)