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Properties

Label 501.2.a
Level $501$
Weight $2$
Character orbit 501.a
Rep. character $\chi_{501}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $5$
Sturm bound $112$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 501 = 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 501.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(112\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(501))\).

Total New Old
Modular forms 58 27 31
Cusp forms 55 27 28
Eisenstein series 3 0 3

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

\(3\)\(167\)FrickeDim
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(9\)
\(-\)\(+\)$-$\(8\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(17\)

Trace form

\( 27 q - 3 q^{2} - q^{3} + 25 q^{4} - 6 q^{5} + q^{6} - 8 q^{7} - 15 q^{8} + 27 q^{9} + O(q^{10}) \) \( 27 q - 3 q^{2} - q^{3} + 25 q^{4} - 6 q^{5} + q^{6} - 8 q^{7} - 15 q^{8} + 27 q^{9} - 2 q^{10} + q^{12} - 2 q^{13} + 8 q^{14} + 2 q^{15} + 33 q^{16} - 2 q^{17} - 3 q^{18} + 4 q^{19} - 6 q^{20} - 8 q^{21} - 8 q^{23} - 3 q^{24} + 9 q^{25} + 22 q^{26} - q^{27} - 20 q^{28} - 10 q^{29} + 10 q^{30} - 8 q^{31} - 23 q^{32} - 4 q^{33} + 2 q^{34} - 4 q^{35} + 25 q^{36} - 14 q^{37} + 20 q^{38} + 2 q^{39} + 42 q^{40} + 10 q^{41} - 4 q^{42} + 16 q^{43} - 6 q^{45} + 8 q^{46} + 12 q^{47} + q^{48} + 11 q^{49} - 13 q^{50} + 2 q^{51} - 26 q^{52} + 2 q^{53} + q^{54} + 24 q^{55} - 4 q^{56} - 12 q^{57} - 46 q^{58} + 32 q^{59} + 6 q^{60} + 2 q^{61} - 20 q^{62} - 8 q^{63} + q^{64} - 16 q^{65} - 20 q^{66} - 8 q^{67} + 46 q^{68} + 4 q^{69} - 8 q^{70} + 36 q^{71} - 15 q^{72} - 34 q^{73} - 6 q^{74} - 7 q^{75} - 20 q^{76} - 2 q^{78} + 24 q^{79} - 42 q^{80} + 27 q^{81} + 38 q^{82} + 36 q^{83} - 16 q^{84} - 24 q^{85} - 76 q^{86} + 10 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} - 8 q^{91} - 60 q^{92} - 32 q^{93} - 8 q^{94} + 32 q^{95} + 25 q^{96} - 46 q^{97} - 59 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 167
501.2.a.a 501.a 1.a $1$ $4.001$ \(\Q\) None \(1\) \(-1\) \(-4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-4q^{5}-q^{6}+4q^{7}+\cdots\)
501.2.a.b 501.a 1.a $5$ $4.001$ 5.5.36497.1 None \(-4\) \(5\) \(-9\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{2}+q^{3}+(1+\beta _{1}-2\beta _{3}+\cdots)q^{4}+\cdots\)
501.2.a.c 501.a 1.a $5$ $4.001$ 5.5.38569.1 None \(0\) \(-5\) \(-1\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(\beta _{2}+\beta _{3})q^{4}+(\beta _{1}+\beta _{4})q^{5}+\cdots\)
501.2.a.d 501.a 1.a $8$ $4.001$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(-8\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}-q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
501.2.a.e 501.a 1.a $8$ $4.001$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(3\) \(8\) \(7\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{3}+\beta _{5}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(501))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(501)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 2}\)