Properties

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

Related objects

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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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 5q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 5q^{7} + q^{8} + q^{9} - 2q^{10} - 5q^{11} - q^{12} + q^{13} - 5q^{14} + 2q^{15} + q^{16} - 7q^{17} + q^{18} - 6q^{19} - 2q^{20} + 5q^{21} - 5q^{22} + 4q^{23} - q^{24} - q^{25} + q^{26} - q^{27} - 5q^{28} + 6q^{29} + 2q^{30} - 6q^{31} + q^{32} + 5q^{33} - 7q^{34} + 10q^{35} + q^{36} - 2q^{37} - 6q^{38} - q^{39} - 2q^{40} - 6q^{41} + 5q^{42} - 6q^{43} - 5q^{44} - 2q^{45} + 4q^{46} - 12q^{47} - q^{48} + 18q^{49} - q^{50} + 7q^{51} + q^{52} - 9q^{53} - q^{54} + 10q^{55} - 5q^{56} + 6q^{57} + 6q^{58} + 2q^{60} - 8q^{61} - 6q^{62} - 5q^{63} + q^{64} - 2q^{65} + 5q^{66} - 13q^{67} - 7q^{68} - 4q^{69} + 10q^{70} - 4q^{71} + q^{72} - q^{73} - 2q^{74} + q^{75} - 6q^{76} + 25q^{77} - q^{78} - 2q^{79} - 2q^{80} + q^{81} - 6q^{82} - 6q^{83} + 5q^{84} + 14q^{85} - 6q^{86} - 6q^{87} - 5q^{88} + 10q^{89} - 2q^{90} - 5q^{91} + 4q^{92} + 6q^{93} - 12q^{94} + 12q^{95} - q^{96} - 2q^{97} + 18q^{98} - 5q^{99} + O(q^{100}) \)

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 −2.00000 −1.00000 −5.00000 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.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

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