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 $4$

Related objects

Downloads

Learn more

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: \(x^{4} - 4 x^{2} + 1\)
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.

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

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
0.517638
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

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

Hecke characteristic polynomials

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