Properties

Label 55.2.a
Level 55
Weight 2
Character orbit a
Rep. character \(\chi_{55}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(55))\).

Total New Old
Modular forms 8 3 5
Cusp forms 5 3 2
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.

\(5\)\(11\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 5q^{16} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 8q^{30} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut 22q^{34} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 13q^{36} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut -\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 16q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 3q^{44} \) \(\mathstrut -\mathstrut 13q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 12q^{47} \) \(\mathstrut -\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 3q^{50} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut +\mathstrut 10q^{53} \) \(\mathstrut -\mathstrut 16q^{54} \) \(\mathstrut -\mathstrut 3q^{55} \) \(\mathstrut -\mathstrut 12q^{56} \) \(\mathstrut -\mathstrut 6q^{58} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 16q^{60} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 8q^{62} \) \(\mathstrut -\mathstrut 20q^{63} \) \(\mathstrut -\mathstrut 7q^{64} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut 8q^{66} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut +\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 39q^{72} \) \(\mathstrut +\mathstrut 6q^{73} \) \(\mathstrut -\mathstrut 22q^{74} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut -\mathstrut 7q^{80} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 14q^{82} \) \(\mathstrut -\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 32q^{84} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 32q^{87} \) \(\mathstrut +\mathstrut 9q^{88} \) \(\mathstrut +\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 13q^{90} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 20q^{92} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 8q^{96} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 13q^{98} \) \(\mathstrut +\mathstrut 13q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(55))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 11
55.2.a.a \(1\) \(0.439\) \(\Q\) None \(1\) \(0\) \(1\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots\)
55.2.a.b \(2\) \(0.439\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-4\) \(+\) \(-\) \(q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(55))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(55)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)