Properties

Label 6400.2.a.cq
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $4$
CM discriminant -8
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 2) q^{9} + ( - \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} - 3) q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{2} + 4 \beta_1) q^{27} + ( - 2 \beta_{3} - 11) q^{33} + ( - 2 \beta_{3} - 3) q^{41} + (3 \beta_{2} - 3 \beta_1) q^{43} - 7 q^{49} + (\beta_{2} - 8 \beta_1) q^{51} + ( - \beta_{3} - 3) q^{57} + ( - 5 \beta_{2} + 5 \beta_1) q^{59} + (5 \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{3} - 1) q^{73} + (\beta_{3} + 13) q^{81} + ( - 5 \beta_{2} - 4 \beta_1) q^{83} + ( - \beta_{3} - 9) q^{89} - 10 q^{97} + (5 \beta_{2} - 15 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{9} - 12 q^{17} - 44 q^{33} - 12 q^{41} - 28 q^{49} - 12 q^{57} - 4 q^{73} + 52 q^{81} - 36 q^{89} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu^{3} + \nu^{2} - 3\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} + 3\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 4 \) Copy content Toggle raw display

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
0.517638
−0.517638
−1.93185
1.93185
0 −3.14626 0 0 0 0 0 6.89898 0
1.2 0 −0.317837 0 0 0 0 0 −2.89898 0
1.3 0 0.317837 0 0 0 0 0 −2.89898 0
1.4 0 3.14626 0 0 0 0 0 6.89898 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 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.cq 4
4.b odd 2 1 inner 6400.2.a.cq 4
5.b even 2 1 6400.2.a.cr 4
8.b even 2 1 inner 6400.2.a.cq 4
8.d odd 2 1 CM 6400.2.a.cq 4
16.e even 4 2 3200.2.d.n 4
16.f odd 4 2 3200.2.d.n 4
20.d odd 2 1 6400.2.a.cr 4
40.e odd 2 1 6400.2.a.cr 4
40.f even 2 1 6400.2.a.cr 4
80.i odd 4 2 3200.2.f.s 8
80.j even 4 2 3200.2.f.s 8
80.k odd 4 2 3200.2.d.q yes 4
80.q even 4 2 3200.2.d.q yes 4
80.s even 4 2 3200.2.f.s 8
80.t odd 4 2 3200.2.f.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.n 4 16.e even 4 2
3200.2.d.n 4 16.f odd 4 2
3200.2.d.q yes 4 80.k odd 4 2
3200.2.d.q yes 4 80.q even 4 2
3200.2.f.s 8 80.i odd 4 2
3200.2.f.s 8 80.j even 4 2
3200.2.f.s 8 80.s even 4 2
3200.2.f.s 8 80.t odd 4 2
6400.2.a.cq 4 1.a even 1 1 trivial
6400.2.a.cq 4 4.b odd 2 1 inner
6400.2.a.cq 4 8.b even 2 1 inner
6400.2.a.cq 4 8.d odd 2 1 CM
6400.2.a.cr 4 5.b even 2 1
6400.2.a.cr 4 20.d odd 2 1
6400.2.a.cr 4 40.e odd 2 1
6400.2.a.cr 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3}^{4} - 10T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{4} - 58T_{11}^{2} + 625 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 15 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 58T^{2} + 625 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 15)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 42T^{2} + 225 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 87)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 330 T^{2} + 16641 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 490 T^{2} + 58081 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 57)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{4} \) Copy content Toggle raw display
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