Properties

Label 422.2.a
Level 422
Weight 2
Character orbit a
Rep. character \(\chi_{422}(1,\cdot)\)
Character field \(\Q\)
Dimension 18
Newforms 6
Sturm bound 106
Trace bound 5

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Defining parameters

Level: \( N \) = \( 422 = 2 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 422.a (trivial)
Character field: \(\Q\)
Newforms: \( 6 \)
Sturm bound: \(106\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(422))\).

Total New Old
Modular forms 55 18 37
Cusp forms 52 18 34
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.

\(2\)\(211\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(11\)

Trace form

\(18q \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 18q^{16} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 2q^{24} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut -\mathstrut 12q^{27} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 20q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 28q^{39} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 28q^{42} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 14q^{45} \) \(\mathstrut -\mathstrut 12q^{46} \) \(\mathstrut -\mathstrut 20q^{47} \) \(\mathstrut +\mathstrut 30q^{49} \) \(\mathstrut -\mathstrut 8q^{50} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut 34q^{55} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 28q^{57} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 16q^{60} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut 8q^{62} \) \(\mathstrut -\mathstrut 16q^{63} \) \(\mathstrut +\mathstrut 18q^{64} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 8q^{68} \) \(\mathstrut -\mathstrut 20q^{69} \) \(\mathstrut -\mathstrut 36q^{70} \) \(\mathstrut -\mathstrut 20q^{71} \) \(\mathstrut -\mathstrut 18q^{73} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut +\mathstrut 32q^{75} \) \(\mathstrut +\mathstrut 10q^{76} \) \(\mathstrut +\mathstrut 16q^{77} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut +\mathstrut 42q^{81} \) \(\mathstrut -\mathstrut 20q^{82} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 4q^{84} \) \(\mathstrut -\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 20q^{86} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 28q^{89} \) \(\mathstrut +\mathstrut 24q^{90} \) \(\mathstrut +\mathstrut 60q^{91} \) \(\mathstrut -\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 44q^{93} \) \(\mathstrut -\mathstrut 12q^{94} \) \(\mathstrut +\mathstrut 46q^{95} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 8q^{98} \) \(\mathstrut -\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(422))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 211
422.2.a.a \(1\) \(3.370\) \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
422.2.a.b \(2\) \(3.370\) \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(2\) \(8\) \(+\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)
422.2.a.c \(3\) \(3.370\) 3.3.257.1 None \(-3\) \(-1\) \(-5\) \(-5\) \(+\) \(+\) \(q-q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(-2+\cdots)q^{5}+\cdots\)
422.2.a.d \(3\) \(3.370\) 3.3.785.1 None \(-3\) \(-1\) \(1\) \(1\) \(+\) \(-\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
422.2.a.e \(3\) \(3.370\) \(\Q(\zeta_{14})^+\) None \(3\) \(-5\) \(-3\) \(-9\) \(-\) \(-\) \(q+q^{2}+(-2-\beta _{2})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
422.2.a.f \(6\) \(3.370\) 6.6.43983893.1 None \(6\) \(4\) \(-2\) \(7\) \(-\) \(+\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{5}q^{5}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(422))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(422)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(211))\)\(^{\oplus 2}\)