Properties

Label 6400.2.a.p
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 4q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - 4q^{7} - 2q^{9} - 3q^{11} + q^{17} - 7q^{19} - 4q^{21} - 4q^{23} - 5q^{27} + 8q^{29} + 4q^{31} - 3q^{33} - 4q^{37} - 3q^{41} + 8q^{43} + 9q^{49} + q^{51} + 12q^{53} - 7q^{57} - 8q^{59} - 4q^{61} + 8q^{63} - 9q^{67} - 4q^{69} + 16q^{71} + 11q^{73} + 12q^{77} + 4q^{79} + q^{81} - q^{83} + 8q^{87} - 13q^{89} + 4q^{93} + 14q^{97} + 6q^{99} + O(q^{100}) \)

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
0 1.00000 0 0 0 −4.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

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 6400.2.a.p 1
4.b odd 2 1 6400.2.a.i 1
5.b even 2 1 6400.2.a.h 1
8.b even 2 1 6400.2.a.c 1
8.d odd 2 1 6400.2.a.v 1
16.e even 4 2 3200.2.d.h yes 2
16.f odd 4 2 3200.2.d.b yes 2
20.d odd 2 1 6400.2.a.q 1
40.e odd 2 1 6400.2.a.b 1
40.f even 2 1 6400.2.a.w 1
80.i odd 4 2 3200.2.f.b 2
80.j even 4 2 3200.2.f.a 2
80.k odd 4 2 3200.2.d.g yes 2
80.q even 4 2 3200.2.d.a 2
80.s even 4 2 3200.2.f.e 2
80.t odd 4 2 3200.2.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.a 2 80.q even 4 2
3200.2.d.b yes 2 16.f odd 4 2
3200.2.d.g yes 2 80.k odd 4 2
3200.2.d.h yes 2 16.e even 4 2
3200.2.f.a 2 80.j even 4 2
3200.2.f.b 2 80.i odd 4 2
3200.2.f.e 2 80.s even 4 2
3200.2.f.f 2 80.t odd 4 2
6400.2.a.b 1 40.e odd 2 1
6400.2.a.c 1 8.b even 2 1
6400.2.a.h 1 5.b even 2 1
6400.2.a.i 1 4.b odd 2 1
6400.2.a.p 1 1.a even 1 1 trivial
6400.2.a.q 1 20.d odd 2 1
6400.2.a.v 1 8.d odd 2 1
6400.2.a.w 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3} - 1 \)
\( T_{7} + 4 \)
\( T_{11} + 3 \)
\( T_{13} \)
\( T_{17} - 1 \)
\( T_{29} - 8 \)
\( T_{31} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( 3 + T \)
$13$ \( T \)
$17$ \( -1 + T \)
$19$ \( 7 + T \)
$23$ \( 4 + T \)
$29$ \( -8 + T \)
$31$ \( -4 + T \)
$37$ \( 4 + T \)
$41$ \( 3 + T \)
$43$ \( -8 + T \)
$47$ \( T \)
$53$ \( -12 + T \)
$59$ \( 8 + T \)
$61$ \( 4 + T \)
$67$ \( 9 + T \)
$71$ \( -16 + T \)
$73$ \( -11 + T \)
$79$ \( -4 + T \)
$83$ \( 1 + T \)
$89$ \( 13 + T \)
$97$ \( -14 + T \)
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