Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 −2.00000 −1.00000 −1.00000 2.00000 −4.00000 3.00000 1.00000 1.00000
\(n\): 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 \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\).