Properties

Label 8619.2.a.i
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8230615021\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 663)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −1.00000 −1.00000 2.00000 −1.00000 0 −3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(1\)
\(17\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8619.2.a.i 1
13.b even 2 1 663.2.a.a 1
39.d odd 2 1 1989.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.a.a 1 13.b even 2 1
1989.2.a.e 1 39.d odd 2 1
8619.2.a.i 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2} - 1 \)
\( T_{5} - 2 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( T \)
$17$ \( -1 + T \)
$19$ \( -4 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( -8 + T \)
$37$ \( -2 + T \)
$41$ \( 2 + T \)
$43$ \( 4 + T \)
$47$ \( 8 + T \)
$53$ \( 10 + T \)
$59$ \( 4 + T \)
$61$ \( -14 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( -14 + T \)
$79$ \( 8 + T \)
$83$ \( -4 + T \)
$89$ \( -6 + T \)
$97$ \( -6 + T \)
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