Properties

Label 6400.2.a.bd
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $2$

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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1600)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( 4 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( 4 - 2 \beta ) q^{9} + ( 3 - \beta ) q^{11} + ( -3 - 2 \beta ) q^{17} + ( 1 + 3 \beta ) q^{19} + ( -13 + 3 \beta ) q^{27} + ( -9 + 4 \beta ) q^{33} + ( -3 - 4 \beta ) q^{41} -10 q^{43} -7 q^{49} + ( -9 - \beta ) q^{51} + ( 17 - 2 \beta ) q^{57} -6 q^{59} + ( 7 + 3 \beta ) q^{67} + ( -1 + 6 \beta ) q^{73} + ( 19 - 10 \beta ) q^{81} + ( -9 - \beta ) q^{83} + ( 9 + 2 \beta ) q^{89} -10 q^{97} + ( 24 - 10 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 8q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 8q^{9} + 6q^{11} - 6q^{17} + 2q^{19} - 26q^{27} - 18q^{33} - 6q^{41} - 20q^{43} - 14q^{49} - 18q^{51} + 34q^{57} - 12q^{59} + 14q^{67} - 2q^{73} + 38q^{81} - 18q^{83} + 18q^{89} - 20q^{97} + 48q^{99} + O(q^{100}) \)

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
−2.44949
2.44949
0 −3.44949 0 0 0 0 0 8.89898 0
1.2 0 1.44949 0 0 0 0 0 −0.898979 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 \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.bd 2
4.b odd 2 1 6400.2.a.ch 2
5.b even 2 1 6400.2.a.ci 2
8.b even 2 1 6400.2.a.ch 2
8.d odd 2 1 CM 6400.2.a.bd 2
16.e even 4 2 1600.2.d.c 4
16.f odd 4 2 1600.2.d.c 4
20.d odd 2 1 6400.2.a.bc 2
40.e odd 2 1 6400.2.a.ci 2
40.f even 2 1 6400.2.a.bc 2
80.i odd 4 2 1600.2.f.j 4
80.j even 4 2 1600.2.f.j 4
80.k odd 4 2 1600.2.d.d yes 4
80.q even 4 2 1600.2.d.d yes 4
80.s even 4 2 1600.2.f.f 4
80.t odd 4 2 1600.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c 4 16.e even 4 2
1600.2.d.c 4 16.f odd 4 2
1600.2.d.d yes 4 80.k odd 4 2
1600.2.d.d yes 4 80.q even 4 2
1600.2.f.f 4 80.s even 4 2
1600.2.f.f 4 80.t odd 4 2
1600.2.f.j 4 80.i odd 4 2
1600.2.f.j 4 80.j even 4 2
6400.2.a.bc 2 20.d odd 2 1
6400.2.a.bc 2 40.f even 2 1
6400.2.a.bd 2 1.a even 1 1 trivial
6400.2.a.bd 2 8.d odd 2 1 CM
6400.2.a.ch 2 4.b odd 2 1
6400.2.a.ch 2 8.b even 2 1
6400.2.a.ci 2 5.b even 2 1
6400.2.a.ci 2 40.e odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 5 \)
\( T_{7} \)
\( T_{11}^{2} - 6 T_{11} + 3 \)
\( T_{13} \)
\( T_{17}^{2} + 6 T_{17} - 15 \)
\( T_{29} \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 3 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -15 + 6 T + T^{2} \)
$19$ \( -53 - 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( -87 + 6 T + T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( -5 - 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -215 + 2 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 75 + 18 T + T^{2} \)
$89$ \( 57 - 18 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
show more
show less