Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 3 \beta q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + 3 \beta q^{7} - q^{9} -2 \beta q^{11} -6 q^{13} + 6 q^{17} + 6 q^{21} + 3 \beta q^{23} -4 \beta q^{27} + 6 \beta q^{31} -4 q^{33} -6 q^{37} -6 \beta q^{39} + 3 \beta q^{43} + 3 \beta q^{47} + 11 q^{49} + 6 \beta q^{51} -6 q^{53} + 8 \beta q^{59} + 6 q^{61} -3 \beta q^{63} + 9 \beta q^{67} + 6 q^{69} + 6 \beta q^{71} -2 q^{73} -12 q^{77} -5 q^{81} + 7 \beta q^{83} + 6 q^{89} -18 \beta q^{91} + 12 q^{93} + 10 q^{97} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{13} + 12q^{17} + 12q^{21} - 8q^{33} - 12q^{37} + 22q^{49} - 12q^{53} + 12q^{61} + 12q^{69} - 4q^{73} - 24q^{77} - 10q^{81} + 12q^{89} + 24q^{93} + 20q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 0 0 −4.24264 0 −1.00000 0
1.2 0 1.41421 0 0 0 4.24264 0 −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.bk 2
4.b odd 2 1 inner 6400.2.a.bk 2
5.b even 2 1 1280.2.a.g 2
8.b even 2 1 6400.2.a.bp 2
8.d odd 2 1 6400.2.a.bp 2
16.e even 4 2 3200.2.d.v 4
16.f odd 4 2 3200.2.d.v 4
20.d odd 2 1 1280.2.a.g 2
40.e odd 2 1 1280.2.a.i 2
40.f even 2 1 1280.2.a.i 2
80.i odd 4 2 3200.2.f.l 4
80.j even 4 2 3200.2.f.g 4
80.k odd 4 2 640.2.d.b 4
80.q even 4 2 640.2.d.b 4
80.s even 4 2 3200.2.f.l 4
80.t odd 4 2 3200.2.f.g 4
240.t even 4 2 5760.2.k.v 4
240.bm odd 4 2 5760.2.k.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.b 4 80.k odd 4 2
640.2.d.b 4 80.q even 4 2
1280.2.a.g 2 5.b even 2 1
1280.2.a.g 2 20.d odd 2 1
1280.2.a.i 2 40.e odd 2 1
1280.2.a.i 2 40.f even 2 1
3200.2.d.v 4 16.e even 4 2
3200.2.d.v 4 16.f odd 4 2
3200.2.f.g 4 80.j even 4 2
3200.2.f.g 4 80.t odd 4 2
3200.2.f.l 4 80.i odd 4 2
3200.2.f.l 4 80.s even 4 2
5760.2.k.v 4 240.t even 4 2
5760.2.k.v 4 240.bm odd 4 2
6400.2.a.bk 2 1.a even 1 1 trivial
6400.2.a.bk 2 4.b odd 2 1 inner
6400.2.a.bp 2 8.b even 2 1
6400.2.a.bp 2 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{7}^{2} - 18 \)
\( T_{11}^{2} - 8 \)
\( T_{13} + 6 \)
\( T_{17} - 6 \)
\( T_{29} \)
\( T_{31}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -18 + T^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( -18 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( -18 + T^{2} \)
$47$ \( -18 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -162 + T^{2} \)
$71$ \( -72 + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( -98 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( -10 + T )^{2} \)
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