Properties

Label 471.2.a
Level 471
Weight 2
Character orbit a
Rep. character \(\chi_{471}(1,\cdot)\)
Character field \(\Q\)
Dimension 27
Newforms 5
Sturm bound 105
Trace bound 1

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Defining parameters

Level: \( N \) = \( 471 = 3 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 471.a (trivial)
Character field: \(\Q\)
Newforms: \( 5 \)
Sturm bound: \(105\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(471))\).

Total New Old
Modular forms 54 27 27
Cusp forms 51 27 24
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.

\(3\)\(157\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(6\)
Minus space\(-\)\(21\)

Trace form

\( 27q - q^{2} + q^{3} + 25q^{4} + 6q^{5} - 3q^{6} + 3q^{8} + 27q^{9} + O(q^{10}) \) \( 27q - q^{2} + q^{3} + 25q^{4} + 6q^{5} - 3q^{6} + 3q^{8} + 27q^{9} + 10q^{10} + 7q^{12} + 10q^{13} + 4q^{14} - 2q^{15} + 17q^{16} + 14q^{17} - q^{18} + 12q^{19} + 6q^{20} + 4q^{21} - 20q^{22} - 4q^{23} - 15q^{24} + 29q^{25} - 22q^{26} + q^{27} + 6q^{29} + 10q^{30} + 4q^{31} - 5q^{32} - 4q^{33} + 10q^{34} + 12q^{35} + 25q^{36} - 2q^{37} - 12q^{38} + 14q^{39} - 2q^{40} + 22q^{41} - 12q^{42} - 4q^{43} - 24q^{44} + 6q^{45} - 32q^{46} + 8q^{47} + 15q^{48} + 63q^{49} + q^{50} - 2q^{51} + 34q^{52} + 10q^{53} - 3q^{54} + 8q^{55} - 12q^{56} + 4q^{57} - 26q^{58} - 24q^{59} - 26q^{60} + 2q^{61} + 24q^{62} - 19q^{64} + 4q^{65} - 12q^{66} + 16q^{67} - 10q^{68} - 8q^{69} - 80q^{70} - 20q^{71} + 3q^{72} + 14q^{73} + 6q^{74} - q^{75} + 28q^{76} - 14q^{78} - 44q^{79} + 2q^{80} + 27q^{81} + 6q^{82} - 4q^{83} + 20q^{84} + 4q^{85} - 76q^{86} + 2q^{87} - 68q^{88} - 2q^{89} + 10q^{90} - 12q^{91} - 52q^{92} - 40q^{94} - 64q^{95} - 43q^{96} - 26q^{97} - 125q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 157
471.2.a.a \(1\) \(3.761\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(3\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+3q^{7}+\cdots\)
471.2.a.b \(2\) \(3.761\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(-2\) \(-6\) \(-\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
471.2.a.c \(3\) \(3.761\) 3.3.229.1 None \(0\) \(-3\) \(-2\) \(-3\) \(+\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
471.2.a.d \(9\) \(3.761\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(2\) \(-9\) \(8\) \(-2\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
471.2.a.e \(12\) \(3.761\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-1\) \(12\) \(4\) \(8\) \(-\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(471))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(471)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\)\(^{\oplus 2}\)