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Properties

Label 8.9.d
Level $8$
Weight $9$
Character orbit 8.d
Rep. character $\chi_{8}(3,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $2$
Sturm bound $9$
Trace bound $1$

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Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(8, [\chi])\).

Total New Old
Modular forms 9 9 0
Cusp forms 7 7 0
Eisenstein series 2 2 0

Trace form

\( 7 q + 2 q^{2} - 2 q^{3} - 332 q^{4} - 740 q^{6} - 2728 q^{8} + 10933 q^{9} + O(q^{10}) \) \( 7 q + 2 q^{2} - 2 q^{3} - 332 q^{4} - 740 q^{6} - 2728 q^{8} + 10933 q^{9} - 4080 q^{10} + 19774 q^{11} - 18632 q^{12} - 51744 q^{14} + 86032 q^{16} - 38642 q^{17} + 181510 q^{18} - 167554 q^{19} - 216480 q^{20} - 385252 q^{22} + 1148560 q^{24} - 57305 q^{25} + 1765104 q^{26} + 12412 q^{27} - 2600640 q^{28} - 4325280 q^{30} + 4280672 q^{32} + 813212 q^{33} + 4977860 q^{34} + 1989120 q^{35} - 8775908 q^{36} - 9244772 q^{38} + 12694080 q^{40} - 281906 q^{41} + 19064640 q^{42} - 4455106 q^{43} - 16433864 q^{44} - 19226976 q^{46} + 22819168 q^{48} - 6890681 q^{49} + 17356610 q^{50} + 1219580 q^{51} - 17270880 q^{52} - 13876040 q^{54} + 10545024 q^{56} + 5066012 q^{57} + 7354800 q^{58} + 1065406 q^{59} - 5760960 q^{60} - 5304960 q^{62} - 18587072 q^{64} + 27060480 q^{65} - 27461560 q^{66} + 33905918 q^{67} + 32422888 q^{68} + 56582400 q^{70} - 79641080 q^{72} - 21917362 q^{73} - 114255984 q^{74} - 112794050 q^{75} + 133467704 q^{76} + 175397280 q^{78} - 134958720 q^{80} - 57515933 q^{81} - 113782012 q^{82} + 174856318 q^{83} + 216531840 q^{84} + 128648860 q^{86} - 158649712 q^{88} + 46653646 q^{89} - 205943760 q^{90} - 273971712 q^{91} + 97636800 q^{92} + 220009536 q^{94} - 215038400 q^{96} + 47775374 q^{97} - 164856958 q^{98} + 445152506 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(8, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
8.9.d.a 8.d 8.d $1$ $3.259$ \(\Q\) \(\Q(\sqrt{-2}) \) \(16\) \(34\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{4}q^{2}+34q^{3}+2^{8}q^{4}+544q^{6}+\cdots\)
8.9.d.b 8.d 8.d $6$ $3.259$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-14\) \(-36\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+\beta _{1})q^{2}+(-6-\beta _{1}-\beta _{2})q^{3}+\cdots\)