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Properties

Label 588.3.d
Level $588$
Weight $3$
Character orbit 588.d
Rep. character $\chi_{588}(97,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $3$
Sturm bound $336$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(336\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 248 14 234
Cusp forms 200 14 186
Eisenstein series 48 0 48

Trace form

\( 14 q - 42 q^{9} + O(q^{10}) \) \( 14 q - 42 q^{9} - 36 q^{11} + 24 q^{15} - 40 q^{23} - 6 q^{25} - 16 q^{29} + 26 q^{37} + 30 q^{39} - 22 q^{43} - 72 q^{51} - 240 q^{53} + 6 q^{57} + 12 q^{65} + 86 q^{67} + 20 q^{71} - 294 q^{79} + 126 q^{81} + 288 q^{85} + 42 q^{93} - 356 q^{95} + 108 q^{99} + O(q^{100}) \)

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.d.a 588.d 7.b $2$ $16.022$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{6}q^{3}-\zeta_{6}q^{5}-3q^{9}-3q^{11}-4\zeta_{6}q^{13}+\cdots\)
588.3.d.b 588.d 7.b $4$ $16.022$ \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+(2\beta _{1}+\beta _{2})q^{5}-3q^{9}+(-8+\cdots)q^{11}+\cdots\)
588.3.d.c 588.d 7.b $8$ $16.022$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(\beta _{4}+\beta _{5})q^{5}-3q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)

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

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