Properties

Label 8034.2.a.b
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
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: \(1\)
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

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

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 0 1.00000 −3.00000 −1.00000 1.00000 0
\(n\): 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.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5} \)
\( T_{7} + 3 \)

Hecke characteristic polynomials

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