Properties

Label 684.2.a
Level $684$
Weight $2$
Character orbit 684.a
Rep. character $\chi_{684}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $5$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(240\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(684))\).

Total New Old
Modular forms 132 7 125
Cusp forms 109 7 102
Eisenstein series 23 0 23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(19\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7 q - q^{5} + 5 q^{7} + O(q^{10}) \) \( 7 q - q^{5} + 5 q^{7} + q^{11} + 5 q^{17} - q^{19} - 8 q^{23} + 14 q^{25} - 2 q^{29} + 20 q^{31} + 15 q^{35} + 6 q^{37} + 6 q^{41} + 5 q^{43} + 19 q^{47} - 4 q^{49} + 7 q^{55} + 2 q^{59} - 25 q^{61} - 32 q^{65} + 28 q^{67} + 30 q^{71} - 7 q^{73} + 5 q^{77} - 8 q^{79} + 4 q^{83} - 33 q^{85} - 40 q^{89} + 20 q^{91} + q^{95} - 20 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(684))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 19
684.2.a.a 684.a 1.a $1$ $5.462$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-2q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
684.2.a.b 684.a 1.a $1$ $5.462$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}-5q^{11}-4q^{13}+3q^{17}+\cdots\)
684.2.a.c 684.a 1.a $1$ $5.462$ \(\Q\) None \(0\) \(0\) \(3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}+q^{7}+5q^{11}-6q^{13}+5q^{17}+\cdots\)
684.2.a.d 684.a 1.a $2$ $5.462$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{5}+(1-\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
684.2.a.e 684.a 1.a $2$ $5.462$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+3q^{7}+\beta q^{11}+2q^{13}-3\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(684))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(684)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)