Properties

Label 51.2.a
Level 51
Weight 2
Character orbit a
Rep. character \(\chi_{51}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 51.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 \(M_{2}(\Gamma_0(51))\).

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.

\(3\)\(17\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q - q^{2} - q^{3} + 3q^{4} + 6q^{5} + q^{6} - 4q^{7} - 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{3} + 3q^{4} + 6q^{5} + q^{6} - 4q^{7} - 9q^{8} + 3q^{9} - 10q^{10} - 4q^{11} - 7q^{12} + 4q^{13} + 7q^{16} + q^{17} - q^{18} + 2q^{19} + 10q^{20} - 4q^{21} - 8q^{22} + 9q^{24} + 7q^{25} + 6q^{26} - q^{27} + 8q^{28} + 6q^{29} + 10q^{30} - 9q^{32} - 2q^{33} - q^{34} - 12q^{35} + 3q^{36} - 6q^{37} + 24q^{38} - 6q^{39} - 22q^{40} - 6q^{41} - 10q^{43} + 12q^{44} + 6q^{45} - 4q^{46} - 20q^{47} + q^{48} - 5q^{49} - 27q^{50} - 3q^{51} + 6q^{52} + 2q^{53} + q^{54} - 2q^{55} - 4q^{57} + 34q^{58} + 12q^{59} - 22q^{60} + 18q^{61} - 16q^{62} - 4q^{63} - q^{64} - 4q^{65} + 8q^{66} + 4q^{67} + 7q^{68} + 18q^{69} + 16q^{71} - 9q^{72} - 6q^{73} + 18q^{74} + q^{75} - 16q^{76} + 12q^{77} - 6q^{78} - 4q^{79} + 42q^{80} + 3q^{81} + 10q^{82} - 16q^{83} + 8q^{84} - 24q^{86} + 6q^{87} - 4q^{88} + 6q^{89} - 10q^{90} + 4q^{91} - 32q^{92} + 4q^{93} + 24q^{94} - 24q^{95} + 9q^{96} - 30q^{97} + 7q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(51))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 17
51.2.a.a \(1\) \(0.407\) \(\Q\) None \(0\) \(1\) \(3\) \(-4\) \(-\) \(+\) \(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b \(2\) \(0.407\) \(\Q(\sqrt{17}) \) None \(-1\) \(-2\) \(3\) \(0\) \(+\) \(-\) \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(51))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(51)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)