Properties

Label 65.2.d.b
Level 65
Weight 2
Character orbit 65.d
Analytic conductor 0.519
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 65 = 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 65.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
1.00000 2.00000i −1.00000 1.00000 + 2.00000i 2.00000i 0 −3.00000 −1.00000 1.00000 + 2.00000i
64.2 1.00000 2.00000i −1.00000 1.00000 2.00000i 2.00000i 0 −3.00000 −1.00000 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
65.d Even 1 yes

Hecke kernels

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