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Properties

Label 588.3.m
Level $588$
Weight $3$
Character orbit 588.m
Rep. character $\chi_{588}(313,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $26$
Newform subspaces $6$
Sturm bound $336$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(588, [\chi])\).

Total New Old
Modular forms 496 26 470
Cusp forms 400 26 374
Eisenstein series 96 0 96

Trace form

\( 26 q + 3 q^{3} + 6 q^{5} + 39 q^{9} + O(q^{10}) \) \( 26 q + 3 q^{3} + 6 q^{5} + 39 q^{9} - 18 q^{11} - 24 q^{15} + 48 q^{17} + 63 q^{19} - 8 q^{23} + 13 q^{25} + 136 q^{29} - 27 q^{31} + 36 q^{33} - 97 q^{37} + 57 q^{39} - 286 q^{43} + 18 q^{45} - 66 q^{47} + 36 q^{51} + 156 q^{53} - 42 q^{57} - 372 q^{59} + 72 q^{61} + 162 q^{65} - 63 q^{67} + 316 q^{71} + 81 q^{73} + 249 q^{75} + 359 q^{79} - 117 q^{81} + 576 q^{85} + 162 q^{87} - 324 q^{89} - 117 q^{93} + 26 q^{95} - 108 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(588, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
588.3.m.a 588.m 7.d $2$ $16.022$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots\)
588.3.m.b 588.m 7.d $2$ $16.022$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{3}+(8-4\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots\)
588.3.m.c 588.m 7.d $2$ $16.022$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(-8+4\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots\)
588.3.m.d 588.m 7.d $4$ $16.022$ \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(6\) \(9\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(3+\cdots)q^{9}+\cdots\)
588.3.m.e 588.m 7.d $8$ $16.022$ 8.0.339738624.1 None \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\beta _{4})q^{3}+(-2\beta _{2}-\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots\)
588.3.m.f 588.m 7.d $8$ $16.022$ 8.0.339738624.1 None \(0\) \(12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{4})q^{3}+(2\beta _{2}+\beta _{5}+\beta _{6})q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(588, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)