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

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

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

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