Properties

Label 69.2.a
Level $69$
Weight $2$
Character orbit 69.a
Rep. character $\chi_{69}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $16$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(69))\).

Total New Old
Modular forms 10 3 7
Cusp forms 7 3 4
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.

\(3\)\(23\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} - q^{3} + 5q^{4} - 2q^{5} + q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - q^{3} + 5q^{4} - 2q^{5} + q^{6} - 3q^{8} + 3q^{9} - 10q^{10} + 12q^{11} - 7q^{12} - 6q^{13} - 12q^{14} + 2q^{15} - 3q^{16} - 6q^{17} + q^{18} + 12q^{19} - 6q^{20} - 4q^{21} + 4q^{22} + q^{23} - 3q^{24} - 3q^{25} + 14q^{26} - q^{27} + 8q^{28} + 2q^{29} + 10q^{30} + 5q^{32} - 4q^{33} - 6q^{34} + 8q^{35} + 5q^{36} + 2q^{37} - 8q^{38} - 6q^{39} - 10q^{40} - 2q^{41} + 8q^{42} + 12q^{43} + 20q^{44} - 2q^{45} - q^{46} - 8q^{47} + q^{48} - 5q^{49} + 15q^{50} + 14q^{51} + 6q^{52} - 18q^{53} + q^{54} - 8q^{55} - 4q^{56} - 8q^{57} - 18q^{58} - 4q^{59} + 6q^{60} - 6q^{61} + 24q^{62} - 19q^{64} - 20q^{65} + 4q^{66} - 4q^{67} - 34q^{68} - 3q^{69} - 8q^{71} - 3q^{72} - 18q^{73} - 18q^{74} - 7q^{75} + 28q^{76} - 26q^{78} + 16q^{79} + 2q^{80} + 3q^{81} + 42q^{82} + 20q^{83} - 4q^{84} + 20q^{85} + 40q^{86} + 2q^{87} - 12q^{88} - 14q^{89} - 10q^{90} - 8q^{91} + 7q^{92} + 8q^{93} + 5q^{96} - 2q^{97} - 23q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(69))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 23
69.2.a.a \(1\) \(0.551\) \(\Q\) None \(1\) \(1\) \(0\) \(-2\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-2q^{7}-3q^{8}+\cdots\)
69.2.a.b \(2\) \(0.551\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(2\) \(+\) \(-\) \(q-\beta q^{2}-q^{3}+3q^{4}+(-1+\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(69))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(69)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)