Properties

Label 588.2.a
Level $588$
Weight $2$
Character orbit 588.a
Rep. character $\chi_{588}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $6$
Sturm bound $224$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(224\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(588))\).

Total New Old
Modular forms 136 6 130
Cusp forms 89 6 83
Eisenstein series 47 0 47

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

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(4\)

Trace form

\( 6 q - 4 q^{5} + 6 q^{9} + O(q^{10}) \) \( 6 q - 4 q^{5} + 6 q^{9} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{25} - 8 q^{29} - 8 q^{31} + 8 q^{33} + 14 q^{37} + 6 q^{39} - 12 q^{41} + 22 q^{43} - 4 q^{45} - 24 q^{47} - 32 q^{53} - 8 q^{55} + 18 q^{57} + 8 q^{59} + 4 q^{61} - 28 q^{65} + 10 q^{67} + 8 q^{69} - 20 q^{71} + 12 q^{73} + 16 q^{75} + 10 q^{79} + 6 q^{81} + 16 q^{83} - 24 q^{85} - 8 q^{87} - 12 q^{89} + 10 q^{93} + 20 q^{95} + 12 q^{97} + 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(588))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
588.2.a.a 588.a 1.a $1$ $4.695$ \(\Q\) None 84.2.i.a \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+2q^{11}-3q^{13}+\cdots\)
588.2.a.b 588.a 1.a $1$ $4.695$ \(\Q\) None 588.2.a.b \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+2q^{11}+4q^{13}+\cdots\)
588.2.a.c 588.a 1.a $1$ $4.695$ \(\Q\) None 84.2.a.b \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-6q^{11}-2q^{13}+4q^{19}+\cdots\)
588.2.a.d 588.a 1.a $1$ $4.695$ \(\Q\) None 84.2.a.a \(0\) \(1\) \(-4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}+q^{9}+2q^{11}+6q^{13}+\cdots\)
588.2.a.e 588.a 1.a $1$ $4.695$ \(\Q\) None 588.2.a.b \(0\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}+2q^{11}-4q^{13}+\cdots\)
588.2.a.f 588.a 1.a $1$ $4.695$ \(\Q\) None 84.2.i.a \(0\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}+2q^{11}+3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(588))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(588)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)