Properties

Label 531.8.a
Level $531$
Weight $8$
Character orbit 531.a
Rep. character $\chi_{531}(1,\cdot)$
Character field $\Q$
Dimension $168$
Newform subspaces $8$
Sturm bound $480$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(531))\).

Total New Old
Modular forms 424 168 256
Cusp forms 416 168 248
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
\(+\)\(+\)$+$\(33\)
\(+\)\(-\)$-$\(33\)
\(-\)\(+\)$-$\(48\)
\(-\)\(-\)$+$\(54\)
Plus space\(+\)\(87\)
Minus space\(-\)\(81\)

Trace form

\( 168 q + 6 q^{2} + 10326 q^{4} + 640 q^{5} + 434 q^{7} - 318 q^{8} + O(q^{10}) \) \( 168 q + 6 q^{2} + 10326 q^{4} + 640 q^{5} + 434 q^{7} - 318 q^{8} + 9620 q^{10} - 8566 q^{11} - 17758 q^{13} + 26806 q^{14} + 629086 q^{16} + 18411 q^{17} - 107510 q^{19} - 38980 q^{20} + 238102 q^{22} + 29488 q^{23} + 2458504 q^{25} + 232214 q^{26} - 235026 q^{28} + 192072 q^{29} + 497176 q^{31} - 250158 q^{32} - 472210 q^{34} - 282715 q^{35} + 1370932 q^{37} + 617224 q^{38} + 2384430 q^{40} - 393616 q^{41} - 381520 q^{43} - 952764 q^{44} - 2638624 q^{46} + 583072 q^{47} + 21703244 q^{49} - 4330558 q^{50} - 3595524 q^{52} + 4318020 q^{53} - 606508 q^{55} + 6641754 q^{56} + 2782622 q^{58} + 1232274 q^{59} + 5446560 q^{61} + 5259552 q^{62} + 47093810 q^{64} + 2601776 q^{65} - 3291596 q^{67} + 2481906 q^{68} + 5292132 q^{70} + 6143853 q^{71} + 3521214 q^{73} + 10350594 q^{74} - 40276708 q^{76} - 15916118 q^{77} - 1719126 q^{79} + 14758748 q^{80} - 4219740 q^{82} - 16124718 q^{83} + 15628556 q^{85} - 17155070 q^{86} + 29858658 q^{88} - 14349796 q^{89} - 24918394 q^{91} + 21530760 q^{92} - 15625704 q^{94} + 3393706 q^{95} + 39645368 q^{97} + 12436162 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\) 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
531.8.a.a 531.a 1.a $14$ $165.876$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(9\) \(0\) \(430\) \(-2390\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(41-\beta _{1}+\beta _{2})q^{4}+(29+\cdots)q^{5}+\cdots\)
531.8.a.b 531.a 1.a $16$ $165.876$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(6\) \(0\) \(68\) \(-2343\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(61+\beta _{2})q^{4}+(5-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
531.8.a.c 531.a 1.a $17$ $165.876$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(-2\) \(0\) \(318\) \(3145\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(68+\beta _{1}+\beta _{2})q^{4}+(19-2\beta _{1}+\cdots)q^{5}+\cdots\)
531.8.a.d 531.a 1.a $17$ $165.876$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(32\) \(0\) \(1072\) \(-2407\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(69-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.e 531.a 1.a $18$ $165.876$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-24\) \(0\) \(-678\) \(3081\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(75+\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.f 531.a 1.a $20$ $165.876$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-15\) \(0\) \(-570\) \(1040\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(77-\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.g 531.a 1.a $33$ $165.876$ None \(-24\) \(0\) \(-1000\) \(154\) $+$ $-$ $\mathrm{SU}(2)$
531.8.a.h 531.a 1.a $33$ $165.876$ None \(24\) \(0\) \(1000\) \(154\) $+$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(531))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(531)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)