Properties

Label 684.2.f
Level $684$
Weight $2$
Character orbit 684.f
Rep. character $\chi_{684}(379,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $5$
Sturm bound $240$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 76 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(240\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(684, [\chi])\).

Total New Old
Modular forms 128 52 76
Cusp forms 112 48 64
Eisenstein series 16 4 12

Trace form

\( 48q + 2q^{4} + 4q^{5} + O(q^{10}) \) \( 48q + 2q^{4} + 4q^{5} + 2q^{16} + 8q^{17} + 12q^{20} + 36q^{25} + 6q^{26} + 22q^{28} - 10q^{38} + 32q^{44} - 8q^{49} + 42q^{58} - 36q^{61} + 36q^{62} - 10q^{64} + 26q^{68} - 16q^{73} - 44q^{74} - 24q^{76} - 20q^{77} - 40q^{80} + 20q^{82} - 28q^{85} - 34q^{92} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
684.2.f.a \(4\) \(5.462\) \(\Q(\sqrt{-2}, \sqrt{19})\) \(\Q(\sqrt{-57}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}-2q^{4}+2\beta _{1}q^{8}-\beta _{2}q^{11}+\cdots\)
684.2.f.b \(8\) \(5.462\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(4\) \(0\) \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{2}+\beta _{6}+\cdots)q^{5}+\cdots\)
684.2.f.c \(10\) \(5.462\) 10.0.\(\cdots\).1 None \(-2\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.d \(10\) \(5.462\) 10.0.\(\cdots\).1 None \(2\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.e \(16\) \(5.462\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+(1+\beta _{6})q^{4}-\beta _{11}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

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