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Properties

Label 588.2.q
Level $588$
Weight $2$
Character orbit 588.q
Rep. character $\chi_{588}(85,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $60$
Newform subspaces $2$
Sturm bound $224$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.q (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 49 \)
Character field: \(\Q(\zeta_{7})\)
Newform subspaces: \( 2 \)
Sturm bound: \(224\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(588, [\chi])\).

Total New Old
Modular forms 708 60 648
Cusp forms 636 60 576
Eisenstein series 72 0 72

Trace form

\( 60 q - 4 q^{5} - 10 q^{9} + O(q^{10}) \) \( 60 q - 4 q^{5} - 10 q^{9} + 18 q^{11} + 4 q^{13} - 10 q^{15} + 18 q^{17} - 20 q^{19} - 2 q^{21} + 8 q^{23} - 14 q^{25} - 8 q^{31} + 8 q^{33} - 24 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} + 8 q^{43} + 10 q^{45} + 46 q^{47} + 26 q^{49} + 20 q^{51} + 28 q^{53} + 34 q^{55} - 4 q^{57} + 8 q^{59} + 18 q^{61} + 20 q^{65} + 4 q^{67} + 8 q^{69} - 20 q^{71} + 54 q^{73} + 16 q^{75} + 50 q^{77} - 12 q^{79} - 10 q^{81} + 2 q^{83} - 56 q^{85} - 22 q^{87} - 26 q^{89} - 92 q^{91} - 12 q^{93} - 100 q^{95} - 212 q^{97} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(588, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
588.2.q.a 588.q 49.e $30$ $4.695$ None \(0\) \(-5\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{7}]$
588.2.q.b 588.q 49.e $30$ $4.695$ None \(0\) \(5\) \(-4\) \(1\) $\mathrm{SU}(2)[C_{7}]$

Decomposition of \(S_{2}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(588, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)