Properties

Label 51.2.a
Level $51$
Weight $2$
Character orbit 51.a
Rep. character $\chi_{51}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
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: \(\Q\)
Newform subspaces: \( 2 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(51))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 17
51.2.a.a 51.a 1.a $1$ $0.407$ \(\Q\) None 51.2.a.a \(0\) \(1\) \(3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b 51.a 1.a $2$ $0.407$ \(\Q(\sqrt{17}) \) None 51.2.a.b \(-1\) \(-2\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)

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

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