Properties

Label 57.2.a
Level 57
Weight 2
Character orbit a
Rep. character \(\chi_{57}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 13
Trace bound 3

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Defining parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(13\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(57))\).

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.

\(3\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 9q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut +\mathstrut 14q^{26} \) \(\mathstrut +\mathstrut q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 21q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 10q^{34} \) \(\mathstrut +\mathstrut 18q^{35} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 16q^{42} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut -\mathstrut 4q^{44} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut 13q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 14q^{52} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut q^{54} \) \(\mathstrut -\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut +\mathstrut 26q^{58} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 16q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 9q^{64} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 10q^{68} \) \(\mathstrut +\mathstrut 12q^{69} \) \(\mathstrut -\mathstrut 36q^{70} \) \(\mathstrut -\mathstrut 3q^{72} \) \(\mathstrut -\mathstrut 12q^{73} \) \(\mathstrut -\mathstrut 26q^{74} \) \(\mathstrut -\mathstrut 9q^{75} \) \(\mathstrut -\mathstrut 3q^{76} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut +\mathstrut 22q^{78} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 10q^{80} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut +\mathstrut 14q^{82} \) \(\mathstrut +\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 16q^{84} \) \(\mathstrut +\mathstrut 18q^{85} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 2q^{89} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 28q^{91} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut +\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 24q^{94} \) \(\mathstrut +\mathstrut 4q^{95} \) \(\mathstrut +\mathstrut 5q^{96} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 47q^{98} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 19
57.2.a.a \(1\) \(0.455\) \(\Q\) None \(-2\) \(-1\) \(-3\) \(-5\) \(+\) \(+\) \(q-2q^{2}-q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots\)
57.2.a.b \(1\) \(0.455\) \(\Q\) None \(-2\) \(1\) \(1\) \(3\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+3q^{7}+\cdots\)
57.2.a.c \(1\) \(0.455\) \(\Q\) None \(1\) \(1\) \(-2\) \(0\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(57))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(57)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)