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Properties

Label 331.1.b
Level $331$
Weight $1$
Character orbit 331.b
Rep. character $\chi_{331}(330,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $27$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 331 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 331.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 331 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(27\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(331, [\chi])\).

Total New Old
Modular forms 4 4 0
Cusp forms 3 3 0
Eisenstein series 1 1 0

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 1 0 2 0

Trace form

\( 3 q - q^{4} - 3 q^{5} - 4 q^{6} - q^{9} + O(q^{10}) \) \( 3 q - q^{4} - 3 q^{5} - 4 q^{6} - q^{9} + 4 q^{14} - q^{16} + q^{17} + q^{19} + q^{20} + 4 q^{21} - 4 q^{22} + 4 q^{30} + q^{31} - 4 q^{33} + 3 q^{36} - 3 q^{43} + q^{45} - q^{49} - 3 q^{53} + 4 q^{58} + 3 q^{64} + q^{67} - 3 q^{68} - 4 q^{70} + q^{71} - 4 q^{74} - 3 q^{76} + 4 q^{77} + q^{79} + q^{80} - q^{81} + 4 q^{82} + 2 q^{83} - 4 q^{84} - q^{85} + 4 q^{87} + 2 q^{89} - q^{95} + 4 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(331, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
331.1.b.a 331.b 331.b $1$ $0.165$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-331}) \) None \(0\) \(0\) \(-1\) \(0\) \(q+q^{4}-q^{5}+q^{9}+q^{16}-q^{17}-q^{19}+\cdots\)
331.1.b.b 331.b 331.b $2$ $0.165$ \(\Q(\sqrt{-2}) \) $S_{4}$ None None \(0\) \(0\) \(-2\) \(0\) \(q-\beta q^{2}-\beta q^{3}-q^{4}-q^{5}-2q^{6}+\beta q^{7}+\cdots\)