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Properties

Label 588.3.p
Level $588$
Weight $3$
Character orbit 588.p
Rep. character $\chi_{588}(557,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $54$
Newform subspaces $9$
Sturm bound $336$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.p (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 9 \)
Sturm bound: \(336\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 496 54 442
Cusp forms 400 54 346
Eisenstein series 96 0 96

Trace form

\( 54 q - 12 q^{9} + O(q^{10}) \) \( 54 q - 12 q^{9} - 30 q^{13} - 14 q^{15} + 25 q^{19} + 109 q^{25} + 90 q^{27} + 75 q^{31} - 5 q^{33} + 103 q^{37} + 143 q^{39} - 26 q^{43} - 155 q^{45} - 217 q^{51} - 320 q^{55} - 264 q^{57} - 60 q^{61} + 59 q^{67} + 370 q^{69} + 305 q^{73} - 145 q^{75} + 145 q^{79} + 200 q^{81} + 812 q^{85} - 490 q^{87} - 464 q^{93} - 280 q^{97} - 602 q^{99} + O(q^{100}) \)

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.p.a 588.p 21.h $2$ $16.022$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-3\zeta_{6}q^{3}+(-9+9\zeta_{6})q^{9}+q^{13}+\cdots\)
588.3.p.b 588.p 21.h $2$ $16.022$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-3\zeta_{6}q^{3}+(-9+9\zeta_{6})q^{9}+22q^{13}+\cdots\)
588.3.p.c 588.p 21.h $2$ $16.022$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{3}+(-9+9\zeta_{6})q^{9}-22q^{13}+\cdots\)
588.3.p.d 588.p 21.h $4$ $16.022$ \(\Q(\sqrt{-3}, \sqrt{-35})\) None \(0\) \(-1\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{3}+(-1+2\beta _{1}+\beta _{2})q^{5}+(9+\cdots)q^{9}+\cdots\)
588.3.p.e 588.p 21.h $4$ $16.022$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+2\beta _{2}+\beta _{3})q^{3}+3\beta _{1}q^{5}+\cdots\)
588.3.p.f 588.p 21.h $8$ $16.022$ 8.0.\(\cdots\).3 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{1}+2\beta _{2}+\beta _{5})q^{3}+(-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
588.3.p.g 588.p 21.h $8$ $16.022$ 8.0.\(\cdots\).87 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{6}q^{3}+(\beta _{1}+\beta _{4}-\beta _{6})q^{5}+(-2+\cdots)q^{9}+\cdots\)
588.3.p.h 588.p 21.h $8$ $16.022$ 8.0.\(\cdots\).3 None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\beta _{2}+\beta _{3})q^{3}+(-\beta _{1}+2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
588.3.p.i 588.p 21.h $16$ $16.022$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{10}-\beta _{12})q^{3}+\beta _{15}q^{5}+(-1+\cdots)q^{9}+\cdots\)

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

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