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Properties

Label 9.3.d
Level $9$
Weight $3$
Character orbit 9.d
Rep. character $\chi_{9}(2,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $2$
Newform subspaces $1$
Sturm bound $3$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 9.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(9, [\chi])\).

Total New Old
Modular forms 6 6 0
Cusp forms 2 2 0
Eisenstein series 4 4 0

Trace form

\( 2 q - 3 q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 9 q^{6} - 2 q^{7} - 9 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 9 q^{6} - 2 q^{7} - 9 q^{9} - 12 q^{10} - 3 q^{11} + 6 q^{12} + 4 q^{13} + 6 q^{14} + 11 q^{16} + 22 q^{19} - 6 q^{20} - 6 q^{21} + 3 q^{22} - 48 q^{23} - 45 q^{24} - 13 q^{25} + 54 q^{27} + 4 q^{28} + 78 q^{29} + 18 q^{30} - 32 q^{31} + 27 q^{32} + 9 q^{33} - 27 q^{34} - 9 q^{36} - 68 q^{37} - 33 q^{38} - 24 q^{39} + 30 q^{40} - 21 q^{41} + 61 q^{43} - 54 q^{45} + 96 q^{46} - 84 q^{47} + 33 q^{48} + 45 q^{49} + 39 q^{50} + 81 q^{51} + 4 q^{52} - 81 q^{54} - 12 q^{55} - 30 q^{56} - 33 q^{57} - 78 q^{58} + 87 q^{59} + 18 q^{60} - 56 q^{61} + 36 q^{63} - 142 q^{64} + 24 q^{65} - 18 q^{66} + 31 q^{67} - 27 q^{68} + 12 q^{70} + 135 q^{72} + 130 q^{73} + 102 q^{74} - 39 q^{75} - 11 q^{76} + 6 q^{77} + 36 q^{78} - 38 q^{79} - 81 q^{81} + 42 q^{82} - 84 q^{83} - 6 q^{84} - 54 q^{85} - 183 q^{86} - 234 q^{87} + 15 q^{88} + 54 q^{90} - 16 q^{91} + 48 q^{92} + 192 q^{93} + 84 q^{94} + 66 q^{95} + 115 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(9, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
9.3.d.a 9.d 9.d $2$ $0.245$ \(\Q(\sqrt{-3}) \) None \(-3\) \(-3\) \(6\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)