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Properties

Label 588.2.i
Level $588$
Weight $2$
Character orbit 588.i
Rep. character $\chi_{588}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $14$
Newform subspaces $7$
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.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(224\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 272 14 258
Cusp forms 176 14 162
Eisenstein series 96 0 96

Trace form

\( 14 q - q^{3} - 2 q^{5} - 7 q^{9} + O(q^{10}) \) \( 14 q - q^{3} - 2 q^{5} - 7 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} + 8 q^{17} - q^{19} + 12 q^{23} - 9 q^{25} + 2 q^{27} - 16 q^{29} + 3 q^{31} - 2 q^{33} - 19 q^{37} - 11 q^{39} - 12 q^{41} + 6 q^{43} - 2 q^{45} + 6 q^{47} - 12 q^{51} + 32 q^{53} + 8 q^{55} + 34 q^{57} + 4 q^{59} - 6 q^{61} + 58 q^{65} + 3 q^{67} + 16 q^{69} + 20 q^{71} - 11 q^{73} + q^{75} - 21 q^{79} - 7 q^{81} - 4 q^{83} - 48 q^{85} - 4 q^{87} - 21 q^{93} - 2 q^{95} - 20 q^{97} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

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

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

\( S_{2}^{\mathrm{old}}(588, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)