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

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

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 59
531.8.a.a \(14\) \(165.876\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(9\) \(0\) \(430\) \(-2390\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(41-\beta _{1}+\beta _{2})q^{4}+(29+\cdots)q^{5}+\cdots\)
531.8.a.b \(16\) \(165.876\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(6\) \(0\) \(68\) \(-2343\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(61+\beta _{2})q^{4}+(5-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
531.8.a.c \(17\) \(165.876\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(-2\) \(0\) \(318\) \(3145\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(68+\beta _{1}+\beta _{2})q^{4}+(19-2\beta _{1}+\cdots)q^{5}+\cdots\)
531.8.a.d \(17\) \(165.876\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(32\) \(0\) \(1072\) \(-2407\) \(-\) \(-\) \(q+(2-\beta _{1})q^{2}+(69-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.e \(18\) \(165.876\) \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-24\) \(0\) \(-678\) \(3081\) \(-\) \(+\) \(q+(-1-\beta _{1})q^{2}+(75+\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.f \(20\) \(165.876\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-15\) \(0\) \(-570\) \(1040\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(77-\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.8.a.g \(33\) \(165.876\) None \(-24\) \(0\) \(-1000\) \(154\) \(+\) \(-\)
531.8.a.h \(33\) \(165.876\) None \(24\) \(0\) \(1000\) \(154\) \(+\) \(+\)

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

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