Properties

Label 8001.2.a.a
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 2
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{4} - 3q^{5} - q^{7} + O(q^{10}) \) \( q - 2q^{2} + 2q^{4} - 3q^{5} - q^{7} + 6q^{10} - 2q^{11} + 5q^{13} + 2q^{14} - 4q^{16} - 6q^{17} - 8q^{19} - 6q^{20} + 4q^{22} - 5q^{23} + 4q^{25} - 10q^{26} - 2q^{28} + 3q^{29} - q^{31} + 8q^{32} + 12q^{34} + 3q^{35} - 11q^{37} + 16q^{38} - 6q^{41} - 8q^{43} - 4q^{44} + 10q^{46} + 2q^{47} + q^{49} - 8q^{50} + 10q^{52} - 3q^{53} + 6q^{55} - 6q^{58} + 3q^{59} - q^{61} + 2q^{62} - 8q^{64} - 15q^{65} + 2q^{67} - 12q^{68} - 6q^{70} - 10q^{71} + 11q^{73} + 22q^{74} - 16q^{76} + 2q^{77} + 12q^{80} + 12q^{82} - 17q^{83} + 18q^{85} + 16q^{86} + q^{89} - 5q^{91} - 10q^{92} - 4q^{94} + 24q^{95} - 6q^{97} - 2q^{98} + O(q^{100}) \)

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
−2.00000 0 2.00000 −3.00000 0 −1.00000 0 0 6.00000
\(n\): 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 8001.2.a.a 1
3.b odd 2 1 2667.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.e 1 3.b odd 2 1
8001.2.a.a 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2} + 2 \)
\( T_{5} + 3 \)

Hecke characteristic polynomials

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