Properties

Label 6400.2.a.cu
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $4$
CM no
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{1} q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{1} q^{7} + 3 q^{9} + 2 q^{11} -4 \beta_{1} q^{13} + 2 \beta_{2} q^{17} + 6 q^{19} -\beta_{3} q^{21} -5 \beta_{1} q^{23} + 2 \beta_{3} q^{29} + 2 \beta_{3} q^{31} + 2 \beta_{2} q^{33} + 2 \beta_{1} q^{37} -4 \beta_{3} q^{39} + 4 q^{41} -\beta_{2} q^{43} + 3 \beta_{1} q^{47} -5 q^{49} + 12 q^{51} + 6 \beta_{2} q^{57} + 2 q^{59} + \beta_{3} q^{61} -3 \beta_{1} q^{63} + \beta_{2} q^{67} -5 \beta_{3} q^{69} + 2 \beta_{3} q^{71} + 2 \beta_{2} q^{73} -2 \beta_{1} q^{77} + 2 \beta_{3} q^{79} -9 q^{81} -5 \beta_{2} q^{83} + 12 \beta_{1} q^{87} -2 q^{89} + 8 q^{91} + 12 \beta_{1} q^{93} + 6 \beta_{2} q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 8q^{11} + 24q^{19} + 16q^{41} - 20q^{49} + 48q^{51} + 8q^{59} - 36q^{81} - 8q^{89} + 32q^{91} + 24q^{99} + O(q^{100}) \)

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
−0.517638
−1.93185
1.93185
0.517638
0 −2.44949 0 0 0 −1.41421 0 3.00000 0
1.2 0 −2.44949 0 0 0 1.41421 0 3.00000 0
1.3 0 2.44949 0 0 0 −1.41421 0 3.00000 0
1.4 0 2.44949 0 0 0 1.41421 0 3.00000 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
5.b even 2 1 inner
8.d odd 2 1 inner
40.e 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.cu 4
4.b odd 2 1 6400.2.a.ct 4
5.b even 2 1 inner 6400.2.a.cu 4
5.c odd 4 2 1280.2.c.h 4
8.b even 2 1 6400.2.a.ct 4
8.d odd 2 1 inner 6400.2.a.cu 4
16.e even 4 2 1600.2.d.i 8
16.f odd 4 2 1600.2.d.i 8
20.d odd 2 1 6400.2.a.ct 4
20.e even 4 2 1280.2.c.g 4
40.e odd 2 1 inner 6400.2.a.cu 4
40.f even 2 1 6400.2.a.ct 4
40.i odd 4 2 1280.2.c.g 4
40.k even 4 2 1280.2.c.h 4
80.i odd 4 2 320.2.f.b 8
80.j even 4 2 320.2.f.b 8
80.k odd 4 2 1600.2.d.i 8
80.q even 4 2 1600.2.d.i 8
80.s even 4 2 320.2.f.b 8
80.t odd 4 2 320.2.f.b 8
240.z odd 4 2 2880.2.d.g 8
240.bb even 4 2 2880.2.d.g 8
240.bd odd 4 2 2880.2.d.g 8
240.bf even 4 2 2880.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 80.i odd 4 2
320.2.f.b 8 80.j even 4 2
320.2.f.b 8 80.s even 4 2
320.2.f.b 8 80.t odd 4 2
1280.2.c.g 4 20.e even 4 2
1280.2.c.g 4 40.i odd 4 2
1280.2.c.h 4 5.c odd 4 2
1280.2.c.h 4 40.k even 4 2
1600.2.d.i 8 16.e even 4 2
1600.2.d.i 8 16.f odd 4 2
1600.2.d.i 8 80.k odd 4 2
1600.2.d.i 8 80.q even 4 2
2880.2.d.g 8 240.z odd 4 2
2880.2.d.g 8 240.bb even 4 2
2880.2.d.g 8 240.bd odd 4 2
2880.2.d.g 8 240.bf even 4 2
6400.2.a.ct 4 4.b odd 2 1
6400.2.a.ct 4 8.b even 2 1
6400.2.a.ct 4 20.d odd 2 1
6400.2.a.ct 4 40.f even 2 1
6400.2.a.cu 4 1.a even 1 1 trivial
6400.2.a.cu 4 5.b even 2 1 inner
6400.2.a.cu 4 8.d odd 2 1 inner
6400.2.a.cu 4 40.e odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 6 \)
\( T_{7}^{2} - 2 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 24 \)
\( T_{29}^{2} - 48 \)
\( T_{31}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -6 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( -2 + T^{2} )^{2} \)
$11$ \( ( -2 + T )^{4} \)
$13$ \( ( -32 + T^{2} )^{2} \)
$17$ \( ( -24 + T^{2} )^{2} \)
$19$ \( ( -6 + T )^{4} \)
$23$ \( ( -50 + T^{2} )^{2} \)
$29$ \( ( -48 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( -8 + T^{2} )^{2} \)
$41$ \( ( -4 + T )^{4} \)
$43$ \( ( -6 + T^{2} )^{2} \)
$47$ \( ( -18 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( -2 + T )^{4} \)
$61$ \( ( -12 + T^{2} )^{2} \)
$67$ \( ( -6 + T^{2} )^{2} \)
$71$ \( ( -48 + T^{2} )^{2} \)
$73$ \( ( -24 + T^{2} )^{2} \)
$79$ \( ( -48 + T^{2} )^{2} \)
$83$ \( ( -150 + T^{2} )^{2} \)
$89$ \( ( 2 + T )^{4} \)
$97$ \( ( -216 + T^{2} )^{2} \)
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