Properties

Label 6400.2.a.cs
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{1} q^{7} + 2 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + \beta_{1} q^{7} + 2 q^{9} + \beta_{3} q^{11} + \beta_{2} q^{13} + 5 q^{17} + \beta_{3} q^{19} + \beta_{2} q^{21} -2 \beta_{1} q^{23} -\beta_{3} q^{27} + \beta_{2} q^{29} + 5 q^{33} -\beta_{2} q^{37} + 5 \beta_{1} q^{39} -3 q^{41} + 4 \beta_{3} q^{43} -\beta_{1} q^{47} + q^{49} + 5 \beta_{3} q^{51} -2 \beta_{2} q^{53} + 5 q^{57} + 4 \beta_{3} q^{59} -\beta_{2} q^{61} + 2 \beta_{1} q^{63} -5 \beta_{3} q^{67} -2 \beta_{2} q^{69} + 5 \beta_{1} q^{71} + 15 q^{73} + \beta_{2} q^{77} -5 \beta_{1} q^{79} -11 q^{81} + 3 \beta_{3} q^{83} + 5 \beta_{1} q^{87} - q^{89} + 8 \beta_{3} q^{91} -10 q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} + 20q^{17} + 20q^{33} - 12q^{41} + 4q^{49} + 20q^{57} + 60q^{73} - 44q^{81} - 4q^{89} - 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 2 \beta_{1}\)

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
0.874032
−0.874032
−2.28825
2.28825
0 −2.23607 0 0 0 −2.82843 0 2.00000 0
1.2 0 −2.23607 0 0 0 2.82843 0 2.00000 0
1.3 0 2.23607 0 0 0 −2.82843 0 2.00000 0
1.4 0 2.23607 0 0 0 2.82843 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.cs 4
4.b odd 2 1 inner 6400.2.a.cs 4
5.b even 2 1 6400.2.a.cp 4
8.b even 2 1 inner 6400.2.a.cs 4
8.d odd 2 1 inner 6400.2.a.cs 4
16.e even 4 2 3200.2.d.r yes 4
16.f odd 4 2 3200.2.d.r yes 4
20.d odd 2 1 6400.2.a.cp 4
40.e odd 2 1 6400.2.a.cp 4
40.f even 2 1 6400.2.a.cp 4
80.i odd 4 2 3200.2.f.r 8
80.j even 4 2 3200.2.f.r 8
80.k odd 4 2 3200.2.d.m 4
80.q even 4 2 3200.2.d.m 4
80.s even 4 2 3200.2.f.r 8
80.t odd 4 2 3200.2.f.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 80.k odd 4 2
3200.2.d.m 4 80.q even 4 2
3200.2.d.r yes 4 16.e even 4 2
3200.2.d.r yes 4 16.f odd 4 2
3200.2.f.r 8 80.i odd 4 2
3200.2.f.r 8 80.j even 4 2
3200.2.f.r 8 80.s even 4 2
3200.2.f.r 8 80.t odd 4 2
6400.2.a.cp 4 5.b even 2 1
6400.2.a.cp 4 20.d odd 2 1
6400.2.a.cp 4 40.e odd 2 1
6400.2.a.cp 4 40.f even 2 1
6400.2.a.cs 4 1.a even 1 1 trivial
6400.2.a.cs 4 4.b odd 2 1 inner
6400.2.a.cs 4 8.b even 2 1 inner
6400.2.a.cs 4 8.d odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 5 \)
\( T_{7}^{2} - 8 \)
\( T_{11}^{2} - 5 \)
\( T_{13}^{2} - 40 \)
\( T_{17} - 5 \)
\( T_{29}^{2} - 40 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -5 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( ( -5 + T^{2} )^{2} \)
$13$ \( ( -40 + T^{2} )^{2} \)
$17$ \( ( -5 + T )^{4} \)
$19$ \( ( -5 + T^{2} )^{2} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( -40 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -40 + T^{2} )^{2} \)
$41$ \( ( 3 + T )^{4} \)
$43$ \( ( -80 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( -160 + T^{2} )^{2} \)
$59$ \( ( -80 + T^{2} )^{2} \)
$61$ \( ( -40 + T^{2} )^{2} \)
$67$ \( ( -125 + T^{2} )^{2} \)
$71$ \( ( -200 + T^{2} )^{2} \)
$73$ \( ( -15 + T )^{4} \)
$79$ \( ( -200 + T^{2} )^{2} \)
$83$ \( ( -45 + T^{2} )^{2} \)
$89$ \( ( 1 + T )^{4} \)
$97$ \( ( 10 + T )^{4} \)
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