Properties

Label 6400.2.a.cd
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( -3 - \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( -3 - \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} + 2 \beta q^{11} + 2 \beta q^{13} + 2 \beta q^{17} + 2 q^{19} + ( -6 - 4 \beta ) q^{21} + ( -3 + 3 \beta ) q^{23} + 4 q^{27} + ( 6 - 2 \beta ) q^{31} + ( 6 + 2 \beta ) q^{33} -6 q^{37} + ( 6 + 2 \beta ) q^{39} + ( -6 - 2 \beta ) q^{41} + ( -5 + 3 \beta ) q^{43} + ( -3 + 3 \beta ) q^{47} + ( 5 + 6 \beta ) q^{49} + ( 6 + 2 \beta ) q^{51} -6 \beta q^{53} + ( 2 + 2 \beta ) q^{57} + 6 q^{59} + ( -6 + 4 \beta ) q^{61} + ( -9 - 7 \beta ) q^{63} + ( 5 - 3 \beta ) q^{67} + 6 q^{69} + ( 6 + 6 \beta ) q^{71} + ( -4 + 6 \beta ) q^{73} + ( -6 - 6 \beta ) q^{77} + 12 q^{79} + ( 1 - 2 \beta ) q^{81} + ( 3 - \beta ) q^{83} + ( 6 + 4 \beta ) q^{89} + ( -6 - 6 \beta ) q^{91} + 4 \beta q^{93} + ( -4 - 6 \beta ) q^{97} + ( 12 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 6q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 6q^{7} + 2q^{9} + 4q^{19} - 12q^{21} - 6q^{23} + 8q^{27} + 12q^{31} + 12q^{33} - 12q^{37} + 12q^{39} - 12q^{41} - 10q^{43} - 6q^{47} + 10q^{49} + 12q^{51} + 4q^{57} + 12q^{59} - 12q^{61} - 18q^{63} + 10q^{67} + 12q^{69} + 12q^{71} - 8q^{73} - 12q^{77} + 24q^{79} + 2q^{81} + 6q^{83} + 12q^{89} - 12q^{91} - 8q^{97} + 24q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −0.732051 0 0 0 −1.26795 0 −2.46410 0
1.2 0 2.73205 0 0 0 −4.73205 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.cd 2
4.b odd 2 1 6400.2.a.bf 2
5.b even 2 1 1280.2.a.b 2
8.b even 2 1 6400.2.a.y 2
8.d odd 2 1 6400.2.a.ck 2
16.e even 4 2 1600.2.d.h 4
16.f odd 4 2 1600.2.d.b 4
20.d odd 2 1 1280.2.a.m 2
40.e odd 2 1 1280.2.a.c 2
40.f even 2 1 1280.2.a.p 2
80.i odd 4 2 1600.2.f.e 4
80.j even 4 2 1600.2.f.d 4
80.k odd 4 2 320.2.d.b yes 4
80.q even 4 2 320.2.d.a 4
80.s even 4 2 1600.2.f.h 4
80.t odd 4 2 1600.2.f.i 4
240.t even 4 2 2880.2.k.l 4
240.bm odd 4 2 2880.2.k.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 80.q even 4 2
320.2.d.b yes 4 80.k odd 4 2
1280.2.a.b 2 5.b even 2 1
1280.2.a.c 2 40.e odd 2 1
1280.2.a.m 2 20.d odd 2 1
1280.2.a.p 2 40.f even 2 1
1600.2.d.b 4 16.f odd 4 2
1600.2.d.h 4 16.e even 4 2
1600.2.f.d 4 80.j even 4 2
1600.2.f.e 4 80.i odd 4 2
1600.2.f.h 4 80.s even 4 2
1600.2.f.i 4 80.t odd 4 2
2880.2.k.e 4 240.bm odd 4 2
2880.2.k.l 4 240.t even 4 2
6400.2.a.y 2 8.b even 2 1
6400.2.a.bf 2 4.b odd 2 1
6400.2.a.cd 2 1.a even 1 1 trivial
6400.2.a.ck 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{7}^{2} + 6 T_{7} + 6 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} - 12 \)
\( T_{17}^{2} - 12 \)
\( T_{29} \)
\( T_{31}^{2} - 12 T_{31} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 6 + 6 T + T^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -18 + 6 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 24 - 12 T + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( 24 + 12 T + T^{2} \)
$43$ \( -2 + 10 T + T^{2} \)
$47$ \( -18 + 6 T + T^{2} \)
$53$ \( -108 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( -12 + 12 T + T^{2} \)
$67$ \( -2 - 10 T + T^{2} \)
$71$ \( -72 - 12 T + T^{2} \)
$73$ \( -92 + 8 T + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 6 - 6 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( -92 + 8 T + T^{2} \)
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