Properties

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

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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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{4} - q^{5} - q^{7} + O(q^{10}) \) \( q - 2q^{2} + 2q^{4} - q^{5} - q^{7} + 2q^{10} - q^{13} + 2q^{14} - 4q^{16} - 6q^{17} - 4q^{19} - 2q^{20} - 9q^{23} - 4q^{25} + 2q^{26} - 2q^{28} - 5q^{29} - 3q^{31} + 8q^{32} + 12q^{34} + q^{35} + q^{37} + 8q^{38} - 10q^{41} - 4q^{43} + 18q^{46} - 12q^{47} + q^{49} + 8q^{50} - 2q^{52} - 3q^{53} + 10q^{58} - 3q^{59} - 3q^{61} + 6q^{62} - 8q^{64} + q^{65} + 12q^{67} - 12q^{68} - 2q^{70} - 6q^{71} - 7q^{73} - 2q^{74} - 8q^{76} + 4q^{80} + 20q^{82} + 9q^{83} + 6q^{85} + 8q^{86} - q^{89} + q^{91} - 18q^{92} + 24q^{94} + 4q^{95} + 14q^{97} - 2q^{98} + O(q^{100}) \)

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 −1.00000 0 −1.00000 0 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 8001.2.a.b 1
3.b odd 2 1 8001.2.a.g yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.b 1 1.a even 1 1 trivial
8001.2.a.g yes 1 3.b odd 2 1

Hecke kernels

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

\( T_{2} + 2 \)
\( T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( 6 + T \)
$19$ \( 4 + T \)
$23$ \( 9 + T \)
$29$ \( 5 + T \)
$31$ \( 3 + T \)
$37$ \( -1 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( 12 + T \)
$53$ \( 3 + T \)
$59$ \( 3 + T \)
$61$ \( 3 + T \)
$67$ \( -12 + T \)
$71$ \( 6 + T \)
$73$ \( 7 + T \)
$79$ \( T \)
$83$ \( -9 + T \)
$89$ \( 1 + T \)
$97$ \( -14 + T \)
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