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Properties

Label 624.2.a
Level $624$
Weight $2$
Character orbit 624.a
Rep. character $\chi_{624}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $11$
Sturm bound $224$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(224\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(624))\).

Total New Old
Modular forms 124 12 112
Cusp forms 101 12 89
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.

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(8\)

Trace form

\( 12 q + 4 q^{7} + 12 q^{9} + O(q^{10}) \) \( 12 q + 4 q^{7} + 12 q^{9} + 8 q^{11} + 4 q^{15} - 4 q^{19} - 16 q^{23} + 20 q^{25} - 16 q^{29} + 20 q^{31} + 8 q^{33} - 16 q^{37} + 4 q^{39} + 16 q^{43} + 24 q^{47} + 20 q^{49} + 8 q^{51} - 8 q^{53} - 16 q^{55} - 8 q^{61} + 4 q^{63} + 20 q^{67} - 16 q^{69} - 16 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} + 16 q^{79} + 12 q^{81} - 48 q^{83} - 12 q^{91} + 16 q^{93} - 8 q^{97} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(624))\) 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 13
624.2.a.a 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(-1\) \(-4\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
624.2.a.b 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+q^{13}-6q^{17}+\cdots\)
624.2.a.c 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-6q^{11}-q^{13}+2q^{17}+\cdots\)
624.2.a.d 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(-1\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{9}+q^{13}-2q^{15}+\cdots\)
624.2.a.e 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(-4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
624.2.a.f 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(-2\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-4q^{7}+q^{9}+q^{13}-2q^{15}+\cdots\)
624.2.a.g 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+2q^{11}-q^{13}+\cdots\)
624.2.a.h 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(2\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
624.2.a.i 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
624.2.a.j 624.a 1.a $1$ $4.983$ \(\Q\) None \(0\) \(1\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+q^{9}+2q^{11}-q^{13}+\cdots\)
624.2.a.k 624.a 1.a $2$ $4.983$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(624))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(624)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)