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

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

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

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