Properties

Label 8034.2.a.c
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

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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} - q^{8} + q^{9} - q^{10} + 2q^{11} + q^{12} - q^{13} - 3q^{14} + q^{15} + q^{16} - q^{18} + 4q^{19} + q^{20} + 3q^{21} - 2q^{22} - q^{24} - 4q^{25} + q^{26} + q^{27} + 3q^{28} + 7q^{29} - q^{30} - 4q^{31} - q^{32} + 2q^{33} + 3q^{35} + q^{36} + 2q^{37} - 4q^{38} - q^{39} - q^{40} - 6q^{41} - 3q^{42} + 2q^{43} + 2q^{44} + q^{45} + 13q^{47} + q^{48} + 2q^{49} + 4q^{50} - q^{52} + 2q^{53} - q^{54} + 2q^{55} - 3q^{56} + 4q^{57} - 7q^{58} - 5q^{59} + q^{60} + 4q^{61} + 4q^{62} + 3q^{63} + q^{64} - q^{65} - 2q^{66} - q^{67} - 3q^{70} - q^{72} + 3q^{73} - 2q^{74} - 4q^{75} + 4q^{76} + 6q^{77} + q^{78} - 4q^{79} + q^{80} + q^{81} + 6q^{82} - q^{83} + 3q^{84} - 2q^{86} + 7q^{87} - 2q^{88} - 8q^{89} - q^{90} - 3q^{91} - 4q^{93} - 13q^{94} + 4q^{95} - q^{96} + 14q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

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

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.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.c 1 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5} - 1 \)
\( T_{7} - 3 \)

Hecke Characteristic Polynomials

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