Properties

Label 8002.2.a.b
Level 8002
Weight 2
Character orbit 8002.a
Self dual yes
Analytic conductor 63.896
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8962916974\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{8} - 3q^{9} - 2q^{11} + 4q^{13} + q^{16} - 3q^{18} + q^{19} - 2q^{22} + 3q^{23} - 5q^{25} + 4q^{26} - 4q^{29} + q^{32} - 3q^{36} + 6q^{37} + q^{38} - 8q^{41} - 12q^{43} - 2q^{44} + 3q^{46} - 10q^{47} - 7q^{49} - 5q^{50} + 4q^{52} - 13q^{53} - 4q^{58} + 7q^{59} + 12q^{61} + q^{64} - 16q^{67} + 8q^{71} - 3q^{72} + 4q^{73} + 6q^{74} + q^{76} + 10q^{79} + 9q^{81} - 8q^{82} - 4q^{83} - 12q^{86} - 2q^{88} + 15q^{89} + 3q^{92} - 10q^{94} + 10q^{97} - 7q^{98} + 6q^{99} + O(q^{100}) \)

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 0 1.00000 0 0 0 1.00000 −3.00000 0
\(n\): 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 8002.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8002.2.a.b 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4001\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 - T + 19 T^{2} \)
$23$ \( 1 - 3 T + 23 T^{2} \)
$29$ \( 1 + 4 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 8 T + 41 T^{2} \)
$43$ \( 1 + 12 T + 43 T^{2} \)
$47$ \( 1 + 10 T + 47 T^{2} \)
$53$ \( 1 + 13 T + 53 T^{2} \)
$59$ \( 1 - 7 T + 59 T^{2} \)
$61$ \( 1 - 12 T + 61 T^{2} \)
$67$ \( 1 + 16 T + 67 T^{2} \)
$71$ \( 1 - 8 T + 71 T^{2} \)
$73$ \( 1 - 4 T + 73 T^{2} \)
$79$ \( 1 - 10 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 15 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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