Properties

Label 99.8.f
Level $99$
Weight $8$
Character orbit 99.f
Rep. character $\chi_{99}(37,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $136$
Newform subspaces $4$
Sturm bound $96$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(99, [\chi])\).

Total New Old
Modular forms 352 144 208
Cusp forms 320 136 184
Eisenstein series 32 8 24

Trace form

\( 136 q + 25 q^{2} - 2297 q^{4} + 76 q^{5} - 2628 q^{7} - 1979 q^{8} + O(q^{10}) \) \( 136 q + 25 q^{2} - 2297 q^{4} + 76 q^{5} - 2628 q^{7} - 1979 q^{8} + 12256 q^{10} - 310 q^{11} - 6588 q^{13} - 30050 q^{14} - 82001 q^{16} + 45634 q^{17} - 66132 q^{19} + 92292 q^{20} + 11141 q^{22} + 54780 q^{23} - 363590 q^{25} - 470170 q^{26} + 178866 q^{28} + 526324 q^{29} - 41148 q^{31} - 2161932 q^{32} + 300346 q^{34} + 325788 q^{35} - 150636 q^{37} - 1443830 q^{38} - 281468 q^{40} - 250484 q^{41} - 2231872 q^{43} + 5405714 q^{44} + 1910148 q^{46} - 1317026 q^{47} - 4414690 q^{49} - 4885545 q^{50} - 799830 q^{52} - 2788246 q^{53} - 1590028 q^{55} + 13078644 q^{56} + 4516052 q^{58} + 5286542 q^{59} + 2468196 q^{61} - 13190780 q^{62} - 4595401 q^{64} - 645260 q^{65} - 12119392 q^{67} - 3637910 q^{68} + 4590808 q^{70} + 7524672 q^{71} - 6982212 q^{73} + 36058986 q^{74} - 81010 q^{76} - 16186834 q^{77} - 21826140 q^{79} - 59305366 q^{80} + 19863593 q^{82} - 4171918 q^{83} - 11737540 q^{85} - 29015447 q^{86} + 51620927 q^{88} + 33117112 q^{89} - 5427116 q^{91} + 78141692 q^{92} + 56443718 q^{94} - 47977986 q^{95} + 9302610 q^{97} - 72043924 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(99, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
99.8.f.a 99.f 11.c $24$ $30.926$ None \(-3\) \(0\) \(72\) \(68\) $\mathrm{SU}(2)[C_{5}]$
99.8.f.b 99.f 11.c $28$ $30.926$ None \(6\) \(0\) \(-773\) \(1289\) $\mathrm{SU}(2)[C_{5}]$
99.8.f.c 99.f 11.c $28$ $30.926$ None \(22\) \(0\) \(777\) \(-83\) $\mathrm{SU}(2)[C_{5}]$
99.8.f.d 99.f 11.c $56$ $30.926$ None \(0\) \(0\) \(0\) \(-3902\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{8}^{\mathrm{old}}(99, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(99, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)