Properties

Label 684.1.h
Level $684$
Weight $1$
Character orbit 684.h
Rep. character $\chi_{684}(37,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $120$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(120\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(684, [\chi])\).

Total New Old
Modular forms 21 3 18
Cusp forms 9 3 6
Eisenstein series 12 0 12

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

\( 3 q + q^{5} + q^{7} + O(q^{10}) \) \( 3 q + q^{5} + q^{7} + q^{11} + q^{17} - q^{19} - 2 q^{23} + 4 q^{25} - q^{35} - 3 q^{43} + q^{47} - 5 q^{55} + q^{61} - 3 q^{73} - q^{77} - 2 q^{83} - 5 q^{85} + q^{95} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.1.h.a 684.h 19.b $1$ $0.341$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-19}) \) None 76.1.c.a \(0\) \(0\) \(1\) \(-1\) \(q+q^{5}-q^{7}+q^{11}+q^{17}+q^{19}-2q^{23}+\cdots\)
684.1.h.b 684.h 19.b $2$ $0.341$ \(\Q(\sqrt{3}) \) $D_{6}$ \(\Q(\sqrt{-19}) \) None 684.1.h.b \(0\) \(0\) \(0\) \(2\) \(q-\beta q^{5}+q^{7}+\beta q^{11}+\beta q^{17}-q^{19}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(684, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)