Properties

Label 531.6.a
Level $531$
Weight $6$
Character orbit 531.a
Rep. character $\chi_{531}(1,\cdot)$
Character field $\Q$
Dimension $122$
Newform subspaces $8$
Sturm bound $360$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(360\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(531))\).

Total New Old
Modular forms 304 122 182
Cusp forms 296 122 174
Eisenstein series 8 0 8

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

\(3\)\(59\)FrickeDim
\(+\)\(+\)$+$\(25\)
\(+\)\(-\)$-$\(25\)
\(-\)\(+\)$-$\(39\)
\(-\)\(-\)$+$\(33\)
Plus space\(+\)\(58\)
Minus space\(-\)\(64\)

Trace form

\( 122 q - 6 q^{2} + 1958 q^{4} - 44 q^{5} + 2 q^{7} - 78 q^{8} + O(q^{10}) \) \( 122 q - 6 q^{2} + 1958 q^{4} - 44 q^{5} + 2 q^{7} - 78 q^{8} + 412 q^{10} - 382 q^{11} - 1414 q^{13} + 1158 q^{14} + 31390 q^{16} + 1321 q^{17} + 1938 q^{19} + 3084 q^{20} - 3922 q^{22} - 640 q^{23} + 81838 q^{25} - 1226 q^{26} + 1438 q^{28} + 13660 q^{29} + 552 q^{31} - 3742 q^{32} + 9958 q^{34} + 9321 q^{35} + 13008 q^{37} + 8760 q^{38} - 33234 q^{40} - 48572 q^{41} + 11216 q^{43} + 32100 q^{44} + 21184 q^{46} + 20336 q^{47} + 299302 q^{49} + 130030 q^{50} - 26324 q^{52} - 28736 q^{53} - 65996 q^{55} + 112490 q^{56} + 28966 q^{58} - 20886 q^{59} - 116204 q^{61} - 67008 q^{62} + 316338 q^{64} - 5288 q^{65} - 9276 q^{67} + 45170 q^{68} - 40284 q^{70} + 59805 q^{71} + 89398 q^{73} - 324542 q^{74} + 5948 q^{76} + 214126 q^{77} + 11338 q^{79} - 116324 q^{80} + 286332 q^{82} + 229866 q^{83} - 54980 q^{85} - 73558 q^{86} - 189150 q^{88} + 191960 q^{89} - 70386 q^{91} + 73880 q^{92} - 273912 q^{94} + 92074 q^{95} + 219068 q^{97} - 233946 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 59
531.6.a.a 531.a 1.a $9$ $85.164$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(0\) \(72\) \(-208\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(7-2\beta _{1}+\beta _{2})q^{4}+(8+\cdots)q^{5}+\cdots\)
531.6.a.b 531.a 1.a $11$ $85.164$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(6\) \(0\) \(192\) \(-371\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(14-\beta _{1}+\beta _{2})q^{4}+(17+\cdots)q^{5}+\cdots\)
531.6.a.c 531.a 1.a $12$ $85.164$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-22\) \(0\) \(-158\) \(413\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(17-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.6.a.d 531.a 1.a $12$ $85.164$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(0\) \(-36\) \(-411\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(17+\beta _{2})q^{4}+(-3-\beta _{1}+\cdots)q^{5}+\cdots\)
531.6.a.e 531.a 1.a $13$ $85.164$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(0\) \(14\) \(373\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(19+\beta _{2})q^{4}+(1-\beta _{7})q^{5}+\cdots\)
531.6.a.f 531.a 1.a $15$ $85.164$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-3\) \(0\) \(-128\) \(282\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(20+\beta _{2})q^{4}+(-8-\beta _{1}+\cdots)q^{5}+\cdots\)
531.6.a.g 531.a 1.a $25$ $85.164$ None \(-12\) \(0\) \(-200\) \(-38\) $+$ $+$ $\mathrm{SU}(2)$
531.6.a.h 531.a 1.a $25$ $85.164$ None \(12\) \(0\) \(200\) \(-38\) $+$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(531))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(531)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)