Properties

Label 501.2.a
Level 501
Weight 2
Character orbit a
Rep. character \(\chi_{501}(1,\cdot)\)
Character field \(\Q\)
Dimension 27
Newforms 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: \(\Q\)
Newforms: \( 5 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 167
501.2.a.a \(1\) \(4.001\) \(\Q\) None \(1\) \(-1\) \(-4\) \(4\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-4q^{5}-q^{6}+4q^{7}+\cdots\)
501.2.a.b \(5\) \(4.001\) 5.5.36497.1 None \(-4\) \(5\) \(-9\) \(-4\) \(-\) \(-\) \(q+(-1+\beta _{3})q^{2}+q^{3}+(1+\beta _{1}-2\beta _{3}+\cdots)q^{4}+\cdots\)
501.2.a.c \(5\) \(4.001\) 5.5.38569.1 None \(0\) \(-5\) \(-1\) \(-4\) \(+\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(\beta _{2}+\beta _{3})q^{4}+(\beta _{1}+\beta _{4})q^{5}+\cdots\)
501.2.a.d \(8\) \(4.001\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(-8\) \(1\) \(0\) \(+\) \(-\) \(q-\beta _{3}q^{2}-q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
501.2.a.e \(8\) \(4.001\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(3\) \(8\) \(7\) \(-4\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{3}+\beta _{5}+\beta _{7})q^{4}+\cdots\)

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

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