Properties

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

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

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 −4.00000 1.00000 1.00000 −1.00000 1.00000 4.00000
\(n\): 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.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.a 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5} + 4 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 4 + T \)
$7$ \( -1 + T \)
$11$ \( 3 + T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( 8 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( -4 + T \)
$37$ \( 12 + T \)
$41$ \( -4 + T \)
$43$ \( T \)
$47$ \( 4 + T \)
$53$ \( 9 + T \)
$59$ \( 2 + T \)
$61$ \( -6 + T \)
$67$ \( 3 + T \)
$71$ \( 6 + T \)
$73$ \( 1 + T \)
$79$ \( 6 + T \)
$83$ \( 6 + T \)
$89$ \( 10 + T \)
$97$ \( 8 + T \)
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