Properties

Label 6400.2.a.u
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{7} - 2q^{9} + 5q^{11} - 6q^{13} - 3q^{17} + q^{19} + 2q^{21} - 4q^{23} - 5q^{27} - 6q^{29} + 8q^{31} + 5q^{33} - 2q^{37} - 6q^{39} - 7q^{41} + 4q^{43} - 2q^{47} - 3q^{49} - 3q^{51} - 4q^{53} + q^{57} + 4q^{59} - 10q^{61} - 4q^{63} + 3q^{67} - 4q^{69} + 2q^{71} - q^{73} + 10q^{77} - 10q^{79} + q^{81} - 9q^{83} - 6q^{87} - 5q^{89} - 12q^{91} + 8q^{93} + 2q^{97} - 10q^{99} + O(q^{100}) \)

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 2.00000 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.u 1
4.b odd 2 1 6400.2.a.d 1
5.b even 2 1 6400.2.a.e 1
8.b even 2 1 6400.2.a.f 1
8.d odd 2 1 6400.2.a.s 1
16.e even 4 2 800.2.d.b 2
16.f odd 4 2 200.2.d.d yes 2
20.d odd 2 1 6400.2.a.t 1
40.e odd 2 1 6400.2.a.g 1
40.f even 2 1 6400.2.a.r 1
48.i odd 4 2 7200.2.k.e 2
48.k even 4 2 1800.2.k.b 2
80.i odd 4 2 800.2.f.a 2
80.j even 4 2 200.2.f.a 2
80.k odd 4 2 200.2.d.a 2
80.q even 4 2 800.2.d.c 2
80.s even 4 2 200.2.f.b 2
80.t odd 4 2 800.2.f.b 2
240.t even 4 2 1800.2.k.h 2
240.z odd 4 2 1800.2.d.d 2
240.bb even 4 2 7200.2.d.j 2
240.bd odd 4 2 1800.2.d.f 2
240.bf even 4 2 7200.2.d.c 2
240.bm odd 4 2 7200.2.k.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.a 2 80.k odd 4 2
200.2.d.d yes 2 16.f odd 4 2
200.2.f.a 2 80.j even 4 2
200.2.f.b 2 80.s even 4 2
800.2.d.b 2 16.e even 4 2
800.2.d.c 2 80.q even 4 2
800.2.f.a 2 80.i odd 4 2
800.2.f.b 2 80.t odd 4 2
1800.2.d.d 2 240.z odd 4 2
1800.2.d.f 2 240.bd odd 4 2
1800.2.k.b 2 48.k even 4 2
1800.2.k.h 2 240.t even 4 2
6400.2.a.d 1 4.b odd 2 1
6400.2.a.e 1 5.b even 2 1
6400.2.a.f 1 8.b even 2 1
6400.2.a.g 1 40.e odd 2 1
6400.2.a.r 1 40.f even 2 1
6400.2.a.s 1 8.d odd 2 1
6400.2.a.t 1 20.d odd 2 1
6400.2.a.u 1 1.a even 1 1 trivial
7200.2.d.c 2 240.bf even 4 2
7200.2.d.j 2 240.bb even 4 2
7200.2.k.e 2 48.i odd 4 2
7200.2.k.g 2 240.bm odd 4 2

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{7} - 2 \)
\( T_{11} - 5 \)
\( T_{13} + 6 \)
\( T_{17} + 3 \)
\( T_{29} + 6 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( -5 + T \)
$13$ \( 6 + T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( 4 + T \)
$29$ \( 6 + T \)
$31$ \( -8 + T \)
$37$ \( 2 + T \)
$41$ \( 7 + T \)
$43$ \( -4 + T \)
$47$ \( 2 + T \)
$53$ \( 4 + T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( -3 + T \)
$71$ \( -2 + T \)
$73$ \( 1 + T \)
$79$ \( 10 + T \)
$83$ \( 9 + T \)
$89$ \( 5 + T \)
$97$ \( -2 + T \)
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