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Properties

Label 588.4.a
Level $588$
Weight $4$
Character orbit 588.a
Rep. character $\chi_{588}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $11$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(588))\).

Total New Old
Modular forms 360 21 339
Cusp forms 312 21 291
Eisenstein series 48 0 48

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(10\)

Trace form

\( 21 q - 3 q^{3} - 2 q^{5} + 189 q^{9} + O(q^{10}) \) \( 21 q - 3 q^{3} - 2 q^{5} + 189 q^{9} - 76 q^{11} - 106 q^{13} + 30 q^{15} - 118 q^{17} + 84 q^{19} + 192 q^{23} + 447 q^{25} - 27 q^{27} + 302 q^{29} + 272 q^{31} - 12 q^{33} - 150 q^{37} - 366 q^{39} - 78 q^{41} - 332 q^{43} - 18 q^{45} + 288 q^{47} + 270 q^{51} + 590 q^{53} + 376 q^{55} + 480 q^{57} + 652 q^{59} - 1354 q^{61} - 1172 q^{65} + 1188 q^{67} + 168 q^{69} - 640 q^{71} - 494 q^{73} - 1077 q^{75} - 644 q^{79} + 1701 q^{81} + 1844 q^{83} - 2132 q^{85} + 1182 q^{87} + 1074 q^{89} + 1764 q^{93} - 4208 q^{95} + 1546 q^{97} - 684 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
588.4.a.a 588.a 1.a $1$ $34.693$ \(\Q\) None \(0\) \(-3\) \(-14\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-14q^{5}+9q^{9}+4q^{11}-54q^{13}+\cdots\)
588.4.a.b 588.a 1.a $1$ $34.693$ \(\Q\) None \(0\) \(-3\) \(4\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{5}+9q^{9}-20q^{11}-4q^{13}+\cdots\)
588.4.a.c 588.a 1.a $1$ $34.693$ \(\Q\) None \(0\) \(-3\) \(18\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+18q^{5}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
588.4.a.d 588.a 1.a $1$ $34.693$ \(\Q\) None \(0\) \(3\) \(-6\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-6q^{5}+9q^{9}+6^{2}q^{11}-62q^{13}+\cdots\)
588.4.a.e 588.a 1.a $1$ $34.693$ \(\Q\) None \(0\) \(3\) \(-4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{5}+9q^{9}-20q^{11}+4q^{13}+\cdots\)
588.4.a.f 588.a 1.a $2$ $34.693$ \(\Q(\sqrt{193}) \) None \(0\) \(-6\) \(-11\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-5-\beta )q^{5}+9q^{9}+(1-7\beta )q^{11}+\cdots\)
588.4.a.g 588.a 1.a $2$ $34.693$ \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(-3\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2-\beta )q^{5}+9q^{9}+(-26+\cdots)q^{11}+\cdots\)
588.4.a.h 588.a 1.a $2$ $34.693$ \(\Q(\sqrt{57}) \) None \(0\) \(6\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(2+\beta )q^{5}+9q^{9}+(-26-\beta )q^{11}+\cdots\)
588.4.a.i 588.a 1.a $2$ $34.693$ \(\Q(\sqrt{193}) \) None \(0\) \(6\) \(11\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(6-\beta )q^{5}+9q^{9}+(-6+7\beta )q^{11}+\cdots\)
588.4.a.j 588.a 1.a $4$ $34.693$ 4.4.136768.1 None \(0\) \(-12\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-\beta _{1}q^{5}+9q^{9}+(3\beta _{1}-\beta _{3})q^{11}+\cdots\)
588.4.a.k 588.a 1.a $4$ $34.693$ 4.4.136768.1 None \(0\) \(12\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta _{1}q^{5}+9q^{9}+(3\beta _{1}-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(588))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(588)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)