Properties

Label 392.1.g
Level $392$
Weight $1$
Character orbit 392.g
Rep. character $\chi_{392}(99,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $56$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 392.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(392, [\chi])\).

Total New Old
Modular forms 11 8 3
Cusp forms 3 3 0
Eisenstein series 8 5 3

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 3 0 0 0

Trace form

\( 3q - q^{2} + 3q^{4} - q^{8} + q^{9} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} - q^{8} + q^{9} - 2q^{11} + 3q^{16} - 3q^{18} - 2q^{22} + 3q^{25} - q^{32} + q^{36} - 2q^{43} - 2q^{44} - q^{50} - 4q^{51} - 4q^{57} + 3q^{64} - 2q^{67} - 3q^{72} - q^{81} - 2q^{86} - 2q^{88} + 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(392, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
392.1.g.a \(1\) \(0.196\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{14}) \) \(1\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+q^{8}-q^{9}-2q^{11}+q^{16}+\cdots\)
392.1.g.b \(2\) \(0.196\) \(\Q(\sqrt{2}) \) \(D_{4}\) \(\Q(\sqrt{-2}) \) None \(-2\) \(0\) \(0\) \(0\) \(q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}-q^{8}+q^{9}+\cdots\)