↑

Properties

Label 414.8
Level 414
Weight 8
Dimension 8746
Nonzero newspaces 8
Sturm bound 76032
Trace bound 3

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(76032\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(414))\).

Total New Old
Modular forms 33616 8746 24870
Cusp forms 32912 8746 24166
Eisenstein series 704 0 704

Trace form

\( 8746 q - 16 q^{2} + 78 q^{3} + 640 q^{4} - 864 q^{5} - 2544 q^{6} + 788 q^{7} + 2048 q^{8} - 114 q^{9} + O(q^{10}) \) \( 8746 q - 16 q^{2} + 78 q^{3} + 640 q^{4} - 864 q^{5} - 2544 q^{6} + 788 q^{7} + 2048 q^{8} - 114 q^{9} - 1536 q^{10} - 654 q^{11} - 2548 q^{13} - 45536 q^{14} - 36720 q^{15} + 40960 q^{16} + 200338 q^{17} + 6816 q^{18} + 248434 q^{19} - 53888 q^{20} - 555708 q^{21} - 404608 q^{22} - 248866 q^{23} + 15360 q^{24} + 945122 q^{25} + 843728 q^{26} + 401760 q^{27} + 391552 q^{28} - 1502102 q^{29} - 912384 q^{30} - 1872890 q^{31} - 65536 q^{32} - 1034946 q^{33} + 518448 q^{34} + 3820894 q^{35} + 670080 q^{36} - 595940 q^{37} - 788816 q^{38} - 4907952 q^{39} - 98304 q^{40} - 3682304 q^{41} + 806400 q^{42} + 648706 q^{43} + 3318528 q^{44} + 6903144 q^{45} - 383904 q^{46} - 4295056 q^{47} - 319488 q^{48} + 1422552 q^{49} - 3146032 q^{50} + 3941334 q^{51} - 163072 q^{52} + 793642 q^{53} - 17710640 q^{54} + 11322776 q^{55} + 15975424 q^{56} + 27670074 q^{57} + 9842880 q^{58} - 18863564 q^{59} - 4669184 q^{60} - 73266952 q^{61} - 45322544 q^{62} - 27283588 q^{63} - 8388608 q^{64} + 37809690 q^{65} + 41163136 q^{66} + 56080286 q^{67} + 37309056 q^{68} + 56555792 q^{69} + 57920448 q^{70} + 23651934 q^{71} - 7840768 q^{72} - 60060688 q^{73} - 103456640 q^{74} - 94157294 q^{75} - 1063552 q^{76} - 138724614 q^{77} - 24381120 q^{78} + 6320182 q^{79} + 18309120 q^{80} + 85627606 q^{81} + 139815744 q^{82} + 250960264 q^{83} + 40053760 q^{84} - 64757170 q^{85} - 161505824 q^{86} - 51482772 q^{87} - 10902528 q^{88} - 123135062 q^{89} - 93851136 q^{90} - 26072020 q^{91} + 9717888 q^{92} + 35623800 q^{93} + 26088096 q^{94} + 187404378 q^{95} + 9437184 q^{96} - 18306134 q^{97} + 12296768 q^{98} - 30211092 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(414))\)

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
414.8.a \(\chi_{414}(1, \cdot)\) 414.8.a.a 1 1
414.8.a.b 2
414.8.a.c 3
414.8.a.d 3
414.8.a.e 3
414.8.a.f 3
414.8.a.g 3
414.8.a.h 4
414.8.a.i 4
414.8.a.j 4
414.8.a.k 4
414.8.a.l 4
414.8.a.m 6
414.8.a.n 6
414.8.a.o 8
414.8.a.p 8
414.8.d \(\chi_{414}(413, \cdot)\) 414.8.d.a 56 1
414.8.e \(\chi_{414}(139, \cdot)\) n/a 308 2
414.8.f \(\chi_{414}(137, \cdot)\) n/a 336 2
414.8.i \(\chi_{414}(55, \cdot)\) n/a 700 10
414.8.j \(\chi_{414}(17, \cdot)\) n/a 560 10
414.8.m \(\chi_{414}(13, \cdot)\) n/a 3360 20
414.8.p \(\chi_{414}(5, \cdot)\) n/a 3360 20

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(414))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(414)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(207))\)\(^{\oplus 2}\)