Properties

Label 684.4.a
Level $684$
Weight $4$
Character orbit 684.a
Rep. character $\chi_{684}(1,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $11$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 684.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(684))\).

Total New Old
Modular forms 372 23 349
Cusp forms 348 23 325
Eisenstein series 24 0 24

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
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(13\)
Minus space\(-\)\(10\)

Trace form

\( 23 q - 32 q^{5} - 10 q^{7} + O(q^{10}) \) \( 23 q - 32 q^{5} - 10 q^{7} - 20 q^{11} - 34 q^{13} - 54 q^{17} + 19 q^{19} + 34 q^{23} + 403 q^{25} - 298 q^{29} + 156 q^{31} - 84 q^{35} - 122 q^{37} + 14 q^{41} - 360 q^{43} + 352 q^{47} + 2251 q^{49} + 682 q^{53} + 1940 q^{55} - 156 q^{59} - 452 q^{61} + 656 q^{65} + 628 q^{67} + 628 q^{71} + 650 q^{73} - 414 q^{77} - 204 q^{79} - 516 q^{83} + 1674 q^{85} - 306 q^{89} + 1832 q^{91} + 266 q^{95} + 314 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(684))\) 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
684.4.a.a 684.a 1.a $1$ $40.357$ \(\Q\) None \(0\) \(0\) \(-18\) \(-32\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-18q^{5}-2^{5}q^{7}-46q^{11}-72q^{13}+\cdots\)
684.4.a.b 684.a 1.a $1$ $40.357$ \(\Q\) None \(0\) \(0\) \(-4\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}-12q^{7}-40q^{11}-40q^{13}+\cdots\)
684.4.a.c 684.a 1.a $1$ $40.357$ \(\Q\) None \(0\) \(0\) \(3\) \(-17\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}-17q^{7}+19q^{11}-30q^{13}+\cdots\)
684.4.a.d 684.a 1.a $1$ $40.357$ \(\Q\) None \(0\) \(0\) \(7\) \(21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{5}+21q^{7}+37q^{11}+26q^{13}+\cdots\)
684.4.a.e 684.a 1.a $1$ $40.357$ \(\Q\) None \(0\) \(0\) \(18\) \(-32\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+18q^{5}-2^{5}q^{7}+46q^{11}-72q^{13}+\cdots\)
684.4.a.f 684.a 1.a $2$ $40.357$ \(\Q(\sqrt{897}) \) None \(0\) \(0\) \(-3\) \(-17\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{5}+(-9+\beta )q^{7}+(7+\beta )q^{11}+\cdots\)
684.4.a.g 684.a 1.a $2$ $40.357$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(5\) \(-30\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+5\beta q^{5}+(-17+4\beta )q^{7}+(38-5\beta )q^{11}+\cdots\)
684.4.a.h 684.a 1.a $3$ $40.357$ 3.3.226425.1 None \(0\) \(0\) \(-31\) \(9\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-10-\beta _{1})q^{5}+(3+\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
684.4.a.i 684.a 1.a $3$ $40.357$ 3.3.35529.1 None \(0\) \(0\) \(-9\) \(44\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{2})q^{5}+(15-\beta _{1}-2\beta _{2})q^{7}+\cdots\)
684.4.a.j 684.a 1.a $4$ $40.357$ 4.4.35245836.1 None \(0\) \(0\) \(0\) \(-6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{5}+(-1+\beta _{3})q^{7}+(-\beta _{1}+3\beta _{2}+\cdots)q^{11}+\cdots\)
684.4.a.k 684.a 1.a $4$ $40.357$ 4.4.106721100.3 None \(0\) \(0\) \(0\) \(62\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+(15-\beta _{3})q^{7}+(\beta _{1}-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(684))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(684)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)