Properties

Label 8034.2.a.j
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

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 8034.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( -2 + T \)
$19$ \( -4 + T \)
$23$ \( 8 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( -10 + T \)
$41$ \( -10 + T \)
$43$ \( 4 + T \)
$47$ \( -4 + T \)
$53$ \( -6 + T \)
$59$ \( -12 + T \)
$61$ \( -14 + T \)
$67$ \( -16 + T \)
$71$ \( 4 + T \)
$73$ \( -6 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( 2 + T \)
$97$ \( 14 + T \)
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