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Properties

Label 57.2.a
Level $57$
Weight $2$
Character orbit 57.a
Rep. character $\chi_{57}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 19
57.2.a.a 57.a 1.a $1$ $0.455$ \(\Q\) None \(-2\) \(-1\) \(-3\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots\)
57.2.a.b 57.a 1.a $1$ $0.455$ \(\Q\) None \(-2\) \(1\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+3q^{7}+\cdots\)
57.2.a.c 57.a 1.a $1$ $0.455$ \(\Q\) None \(1\) \(1\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{8}+\cdots\)

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

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