Properties

Label 99.8.e
Level $99$
Weight $8$
Character orbit 99.e
Rep. character $\chi_{99}(34,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $140$
Newform subspaces $2$
Sturm bound $96$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)

Dimensions

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

Total New Old
Modular forms 172 140 32
Cusp forms 164 140 24
Eisenstein series 8 0 8

Trace form

\( 140 q - 13 q^{3} - 4480 q^{4} + 429 q^{5} + 2030 q^{6} + 166 q^{7} - 5196 q^{8} + 2587 q^{9} + O(q^{10}) \) \( 140 q - 13 q^{3} - 4480 q^{4} + 429 q^{5} + 2030 q^{6} + 166 q^{7} - 5196 q^{8} + 2587 q^{9} + 5324 q^{11} - 18134 q^{12} + 3694 q^{13} + 33054 q^{14} + 15032 q^{15} - 286720 q^{16} - 54444 q^{17} - 129308 q^{18} - 44684 q^{19} + 165030 q^{20} + 117596 q^{21} + 100002 q^{23} - 706554 q^{24} - 1129747 q^{25} - 674520 q^{26} - 360094 q^{27} - 84992 q^{28} + 192036 q^{29} + 1323146 q^{30} - 115721 q^{31} + 1606944 q^{32} + 220946 q^{33} + 390444 q^{34} - 2540160 q^{35} - 3112606 q^{36} - 126914 q^{37} + 2653530 q^{38} + 1629272 q^{39} - 799500 q^{40} + 522084 q^{41} - 970018 q^{42} - 96104 q^{43} - 1703680 q^{44} - 4352345 q^{45} + 3368520 q^{46} + 3104142 q^{47} + 10121290 q^{48} - 7877412 q^{49} - 799314 q^{50} + 2347286 q^{51} - 1297910 q^{52} - 6507876 q^{53} - 215020 q^{54} + 1349634 q^{55} + 3634188 q^{56} + 1824786 q^{57} + 2589534 q^{58} + 614817 q^{59} - 2059754 q^{60} + 5446390 q^{61} - 4837380 q^{62} + 747556 q^{63} + 21361268 q^{64} + 1756530 q^{65} + 1850090 q^{66} - 3973847 q^{67} - 4113954 q^{68} + 6167119 q^{69} + 1164000 q^{70} - 7882422 q^{71} + 31233828 q^{72} + 11731804 q^{73} - 13154346 q^{74} - 33588095 q^{75} + 13021348 q^{76} + 3652264 q^{77} + 80390 q^{78} - 2951258 q^{79} - 30127020 q^{80} + 21476095 q^{81} - 22595676 q^{82} - 23021280 q^{83} + 7299034 q^{84} + 605496 q^{85} - 34925976 q^{86} - 44391600 q^{87} - 42894156 q^{89} + 2548888 q^{90} - 6896356 q^{91} - 52931718 q^{92} - 40350403 q^{93} - 20004522 q^{94} - 11712348 q^{95} + 35403472 q^{96} - 32940791 q^{97} + 163751148 q^{98} - 3347465 q^{99} + O(q^{100}) \)

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.e.a 99.e 9.c $66$ $30.926$ None \(0\) \(-48\) \(-39\) \(1455\) $\mathrm{SU}(2)[C_{3}]$
99.8.e.b 99.e 9.c $74$ $30.926$ None \(0\) \(35\) \(468\) \(-1289\) $\mathrm{SU}(2)[C_{3}]$

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

\( S_{8}^{\mathrm{old}}(99, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 2}\)