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Properties

Label 588.4.i
Level $588$
Weight $4$
Character orbit 588.i
Rep. character $\chi_{588}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $12$
Sturm bound $448$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(588, [\chi])\).

Total New Old
Modular forms 720 40 680
Cusp forms 624 40 584
Eisenstein series 96 0 96

Trace form

\( 40 q + 8 q^{5} - 180 q^{9} + O(q^{10}) \) \( 40 q + 8 q^{5} - 180 q^{9} - 56 q^{11} - 132 q^{13} - 84 q^{15} + 124 q^{17} + 236 q^{19} - 204 q^{23} - 710 q^{25} - 440 q^{29} + 270 q^{31} + 138 q^{33} + 218 q^{37} + 252 q^{39} - 24 q^{41} - 1568 q^{43} + 72 q^{45} - 36 q^{47} + 288 q^{51} - 428 q^{53} + 740 q^{55} + 1980 q^{57} - 1096 q^{59} - 204 q^{61} - 1000 q^{65} - 2116 q^{67} - 1608 q^{69} - 728 q^{71} - 146 q^{73} - 312 q^{75} + 946 q^{79} - 1620 q^{81} + 3616 q^{83} - 88 q^{85} + 942 q^{87} - 1152 q^{89} - 258 q^{93} + 1724 q^{95} - 2804 q^{97} + 1008 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(588, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
588.4.i.a 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 84.4.a.b \(0\) \(-3\) \(-14\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.b 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 588.4.a.b \(0\) \(-3\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.c 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 84.4.a.a \(0\) \(-3\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.d 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 12.4.a.a \(0\) \(-3\) \(18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.e 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 12.4.a.a \(0\) \(3\) \(-18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.f 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 84.4.a.a \(0\) \(3\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.g 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 588.4.a.b \(0\) \(3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.h 588.i 7.c $2$ $34.693$ \(\Q(\sqrt{-3}) \) None 84.4.a.b \(0\) \(3\) \(14\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.i 588.i 7.c $4$ $34.693$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None 84.4.i.b \(0\) \(-6\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{1}q^{3}+(-2-2\beta _{1}+\beta _{3})q^{5}+(-9+\cdots)q^{9}+\cdots\)
588.4.i.j 588.i 7.c $4$ $34.693$ \(\Q(\sqrt{-3}, \sqrt{193})\) None 84.4.i.a \(0\) \(6\) \(11\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{2})q^{3}+(-\beta _{1}+6\beta _{2})q^{5}-9\beta _{2}q^{9}+\cdots\)
588.4.i.k 588.i 7.c $8$ $34.693$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 588.4.a.j \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{1}q^{3}+\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots\)
588.4.i.l 588.i 7.c $8$ $34.693$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 588.4.a.j \(0\) \(12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{1}q^{3}-\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(588, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)