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Properties

Label 356.1.d
Level $356$
Weight $1$
Character orbit 356.d
Rep. character $\chi_{356}(355,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $45$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 356 = 2^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 356.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 356 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(45\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(356, [\chi])\).

Total New Old
Modular forms 7 7 0
Cusp forms 5 5 0
Eisenstein series 2 2 0

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 5 0 0 0

Trace form

\( 5 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 3 q^{9} - 2 q^{10} + 5 q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{20} - 4 q^{21} + 3 q^{25} - q^{32} - 2 q^{34} + 3 q^{36} - 2 q^{40} - 4 q^{42} - 6 q^{45} + 3 q^{49} - 3 q^{50} - 2 q^{53} - 4 q^{57} + 5 q^{64} - 2 q^{68} - 4 q^{69} - 3 q^{72} - 2 q^{73} - 2 q^{80} + q^{81} - 4 q^{84} - 4 q^{85} + 5 q^{89} + 6 q^{90} + 8 q^{93} - 2 q^{97} + 9 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(356, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
356.1.d.a 356.d 356.d $1$ $0.178$ \(\Q\) $D_{2}$ \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-89}) \) \(\Q(\sqrt{89}) \) \(-1\) \(0\) \(2\) \(0\) \(q-q^{2}+q^{4}+2q^{5}-q^{8}-q^{9}-2q^{10}+\cdots\)
356.1.d.b 356.d 356.d $1$ $0.178$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-89}) \) None \(1\) \(-1\) \(-1\) \(2\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
356.1.d.c 356.d 356.d $1$ $0.178$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-89}) \) None \(1\) \(1\) \(-1\) \(-2\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
356.1.d.d 356.d 356.d $2$ $0.178$ \(\Q(\sqrt{3}) \) $D_{6}$ \(\Q(\sqrt{-89}) \) None \(-2\) \(0\) \(-2\) \(0\) \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}-q^{8}+\cdots\)