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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
588.4.i.a \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-14\) \(0\) \(q+(-3+3\zeta_{6})q^{3}-14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.b \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(4\) \(0\) \(q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.c \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(6\) \(0\) \(q+(-3+3\zeta_{6})q^{3}+6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.d \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(18\) \(0\) \(q+(-3+3\zeta_{6})q^{3}+18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.e \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-18\) \(0\) \(q+(3-3\zeta_{6})q^{3}-18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.f \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-6\) \(0\) \(q+(3-3\zeta_{6})q^{3}-6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.g \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-4\) \(0\) \(q+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.h \(2\) \(34.693\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(14\) \(0\) \(q+(3-3\zeta_{6})q^{3}+14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
588.4.i.i \(4\) \(34.693\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(-6\) \(-3\) \(0\) \(q+3\beta _{1}q^{3}+(-2-2\beta _{1}+\beta _{3})q^{5}+(-9+\cdots)q^{9}+\cdots\)
588.4.i.j \(4\) \(34.693\) \(\Q(\sqrt{-3}, \sqrt{193})\) None \(0\) \(6\) \(11\) \(0\) \(q+(3-3\beta _{2})q^{3}+(-\beta _{1}+6\beta _{2})q^{5}-9\beta _{2}q^{9}+\cdots\)
588.4.i.k \(8\) \(34.693\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-12\) \(0\) \(0\) \(q-3\beta _{1}q^{3}+\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots\)
588.4.i.l \(8\) \(34.693\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(12\) \(0\) \(0\) \(q+3\beta _{1}q^{3}-\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots\)

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

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