Properties

Label 65.2.a
Level $65$
Weight $2$
Character orbit 65.a
Rep. character $\chi_{65}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $3$
Sturm bound $14$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(14\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(65))\).

Total New Old
Modular forms 8 5 3
Cusp forms 5 5 0
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.

\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8} + q^{9} + O(q^{10}) \) \( 5 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8} + q^{9} - q^{10} - 4 q^{12} - q^{13} - 8 q^{14} - 5 q^{16} - 2 q^{17} - 11 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 4 q^{22} + 4 q^{24} + 5 q^{25} + 3 q^{26} + 12 q^{27} + 28 q^{28} - 10 q^{29} + 8 q^{30} + 12 q^{31} + q^{32} - 20 q^{33} + 6 q^{34} + 4 q^{35} - q^{36} - 10 q^{37} + 24 q^{38} + 4 q^{39} - 9 q^{40} - 18 q^{41} - 4 q^{42} + 12 q^{43} + 4 q^{44} - 5 q^{45} + 8 q^{46} + 12 q^{47} - 8 q^{48} + 13 q^{49} - 3 q^{50} - 24 q^{51} + q^{52} - 10 q^{53} - 20 q^{54} + 8 q^{55} - 32 q^{56} - 4 q^{57} + 2 q^{58} + 12 q^{59} - 12 q^{60} - 10 q^{61} - 8 q^{62} - 4 q^{63} - 5 q^{64} - 3 q^{65} + 40 q^{66} - 16 q^{67} + 10 q^{68} + 8 q^{69} - 16 q^{70} + 16 q^{71} + 21 q^{72} - 14 q^{73} + 26 q^{74} - 4 q^{77} - 12 q^{78} - 8 q^{79} + 17 q^{80} - 19 q^{81} - 2 q^{82} - 40 q^{83} - 28 q^{84} - 6 q^{85} + 36 q^{86} + 12 q^{87} - 16 q^{88} + 2 q^{89} + 15 q^{90} + 4 q^{91} + 20 q^{92} + 60 q^{93} + 8 q^{94} + 12 q^{95} + 32 q^{96} - 2 q^{97} - 51 q^{98} - 20 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 13
65.2.a.a 65.a 1.a $1$ $0.519$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}-q^{4}-q^{5}+2q^{6}-4q^{7}+\cdots\)
65.2.a.b 65.a 1.a $2$ $0.519$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
65.2.a.c 65.a 1.a $2$ $0.519$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1-\beta )q^{3}+q^{4}-q^{5}+(-3+\cdots)q^{6}+\cdots\)