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Properties

Label 65.2.a.b
Level 65
Weight 2
Character orbit 65.a
Self dual Yes
Analytic conductor 0.519
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 65 = 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 65.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + \beta q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( 2 - \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( -3 + \beta ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + \beta q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( 2 - \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( -3 + \beta ) q^{8} - q^{9} + ( -1 + \beta ) q^{10} + ( 2 - \beta ) q^{11} + ( -4 + \beta ) q^{12} - q^{13} + ( -6 + 4 \beta ) q^{14} + \beta q^{15} + 3 q^{16} + ( -2 - 2 \beta ) q^{17} + ( 1 - \beta ) q^{18} + ( 2 + \beta ) q^{19} + ( 1 - 2 \beta ) q^{20} + ( -4 + 2 \beta ) q^{21} + ( -4 + 3 \beta ) q^{22} -\beta q^{23} + ( 2 - 3 \beta ) q^{24} + q^{25} + ( 1 - \beta ) q^{26} -4 \beta q^{27} + ( 10 - 6 \beta ) q^{28} + 4 \beta q^{29} + ( 2 - \beta ) q^{30} + ( 6 + 3 \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( -2 + 2 \beta ) q^{33} -2 q^{34} + ( 2 - 2 \beta ) q^{35} + ( -1 + 2 \beta ) q^{36} + 6 \beta q^{37} + \beta q^{38} -\beta q^{39} + ( -3 + \beta ) q^{40} + ( -6 - 2 \beta ) q^{41} + ( 8 - 6 \beta ) q^{42} + ( -4 + 5 \beta ) q^{43} + ( 6 - 5 \beta ) q^{44} - q^{45} + ( -2 + \beta ) q^{46} + ( -2 + 2 \beta ) q^{47} + 3 \beta q^{48} + ( 5 - 8 \beta ) q^{49} + ( -1 + \beta ) q^{50} + ( -4 - 2 \beta ) q^{51} + ( -1 + 2 \beta ) q^{52} + ( -6 - 6 \beta ) q^{53} + ( -8 + 4 \beta ) q^{54} + ( 2 - \beta ) q^{55} + ( -10 + 8 \beta ) q^{56} + ( 2 + 2 \beta ) q^{57} + ( 8 - 4 \beta ) q^{58} + ( 6 + 3 \beta ) q^{59} + ( -4 + \beta ) q^{60} -8 q^{61} + 3 \beta q^{62} + ( -2 + 2 \beta ) q^{63} + ( -7 + 2 \beta ) q^{64} - q^{65} + ( 6 - 4 \beta ) q^{66} -2 q^{67} + ( 6 + 2 \beta ) q^{68} -2 q^{69} + ( -6 + 4 \beta ) q^{70} + ( 2 - 7 \beta ) q^{71} + ( 3 - \beta ) q^{72} -6 \beta q^{73} + ( 12 - 6 \beta ) q^{74} + \beta q^{75} + ( -2 - 3 \beta ) q^{76} + ( 8 - 6 \beta ) q^{77} + ( -2 + \beta ) q^{78} + 6 \beta q^{79} + 3 q^{80} -5 q^{81} + ( 2 - 4 \beta ) q^{82} + ( -6 - 2 \beta ) q^{83} + ( -12 + 10 \beta ) q^{84} + ( -2 - 2 \beta ) q^{85} + ( 14 - 9 \beta ) q^{86} + 8 q^{87} + ( -8 + 5 \beta ) q^{88} + 6 q^{89} + ( 1 - \beta ) q^{90} + ( -2 + 2 \beta ) q^{91} + ( 4 - \beta ) q^{92} + ( 6 + 6 \beta ) q^{93} + ( 6 - 4 \beta ) q^{94} + ( 2 + \beta ) q^{95} + ( 2 + 3 \beta ) q^{96} + ( -2 + 4 \beta ) q^{97} + ( -21 + 13 \beta ) q^{98} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{6} + 4q^{7} - 6q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{6} + 4q^{7} - 6q^{8} - 2q^{9} - 2q^{10} + 4q^{11} - 8q^{12} - 2q^{13} - 12q^{14} + 6q^{16} - 4q^{17} + 2q^{18} + 4q^{19} + 2q^{20} - 8q^{21} - 8q^{22} + 4q^{24} + 2q^{25} + 2q^{26} + 20q^{28} + 4q^{30} + 12q^{31} + 6q^{32} - 4q^{33} - 4q^{34} + 4q^{35} - 2q^{36} - 6q^{40} - 12q^{41} + 16q^{42} - 8q^{43} + 12q^{44} - 2q^{45} - 4q^{46} - 4q^{47} + 10q^{49} - 2q^{50} - 8q^{51} - 2q^{52} - 12q^{53} - 16q^{54} + 4q^{55} - 20q^{56} + 4q^{57} + 16q^{58} + 12q^{59} - 8q^{60} - 16q^{61} - 4q^{63} - 14q^{64} - 2q^{65} + 12q^{66} - 4q^{67} + 12q^{68} - 4q^{69} - 12q^{70} + 4q^{71} + 6q^{72} + 24q^{74} - 4q^{76} + 16q^{77} - 4q^{78} + 6q^{80} - 10q^{81} + 4q^{82} - 12q^{83} - 24q^{84} - 4q^{85} + 28q^{86} + 16q^{87} - 16q^{88} + 12q^{89} + 2q^{90} - 4q^{91} + 8q^{92} + 12q^{93} + 12q^{94} + 4q^{95} + 4q^{96} - 4q^{97} - 42q^{98} - 4q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 −1.41421 3.82843 1.00000 3.41421 4.82843 −4.41421 −1.00000 −2.41421
1.2 0.414214 1.41421 −1.82843 1.00000 0.585786 −0.828427 −1.58579 −1.00000 0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\).