Properties

Label 6400.2.a.bq
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{3} + 2 q^{9} -\beta q^{11} -4 q^{13} -3 q^{17} -\beta q^{19} + 4 \beta q^{23} + \beta q^{27} + 4 q^{29} + 4 \beta q^{31} + 5 q^{33} -8 q^{37} + 4 \beta q^{39} + 5 q^{41} + 4 \beta q^{43} -4 \beta q^{47} -7 q^{49} + 3 \beta q^{51} -4 q^{53} + 5 q^{57} + 4 \beta q^{59} + 8 q^{61} -3 \beta q^{67} -20 q^{69} + 4 \beta q^{71} -9 q^{73} -11 q^{81} -3 \beta q^{83} -4 \beta q^{87} + 15 q^{89} -20 q^{93} -2 q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 8q^{13} - 6q^{17} + 8q^{29} + 10q^{33} - 16q^{37} + 10q^{41} - 14q^{49} - 8q^{53} + 10q^{57} + 16q^{61} - 40q^{69} - 18q^{73} - 22q^{81} + 30q^{89} - 40q^{93} - 4q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
0 −2.23607 0 0 0 0 0 2.00000 0
1.2 0 2.23607 0 0 0 0 0 2.00000 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.bq 2
4.b odd 2 1 inner 6400.2.a.bq 2
5.b even 2 1 6400.2.a.bt 2
8.b even 2 1 6400.2.a.bs 2
8.d odd 2 1 6400.2.a.bs 2
16.e even 4 2 3200.2.d.o 4
16.f odd 4 2 3200.2.d.o 4
20.d odd 2 1 6400.2.a.bt 2
40.e odd 2 1 6400.2.a.br 2
40.f even 2 1 6400.2.a.br 2
80.i odd 4 2 3200.2.f.n 4
80.j even 4 2 3200.2.f.m 4
80.k odd 4 2 3200.2.d.p yes 4
80.q even 4 2 3200.2.d.p yes 4
80.s even 4 2 3200.2.f.n 4
80.t odd 4 2 3200.2.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.o 4 16.e even 4 2
3200.2.d.o 4 16.f odd 4 2
3200.2.d.p yes 4 80.k odd 4 2
3200.2.d.p yes 4 80.q even 4 2
3200.2.f.m 4 80.j even 4 2
3200.2.f.m 4 80.t odd 4 2
3200.2.f.n 4 80.i odd 4 2
3200.2.f.n 4 80.s even 4 2
6400.2.a.bq 2 1.a even 1 1 trivial
6400.2.a.bq 2 4.b odd 2 1 inner
6400.2.a.br 2 40.e odd 2 1
6400.2.a.br 2 40.f even 2 1
6400.2.a.bs 2 8.b even 2 1
6400.2.a.bs 2 8.d odd 2 1
6400.2.a.bt 2 5.b even 2 1
6400.2.a.bt 2 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3}^{2} - 5 \)
\( T_{7} \)
\( T_{11}^{2} - 5 \)
\( T_{13} + 4 \)
\( T_{17} + 3 \)
\( T_{29} - 4 \)
\( T_{31}^{2} - 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( -5 + T^{2} \)
$23$ \( -80 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( -80 + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( ( -5 + T )^{2} \)
$43$ \( -80 + T^{2} \)
$47$ \( -80 + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( -45 + T^{2} \)
$71$ \( -80 + T^{2} \)
$73$ \( ( 9 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( -45 + T^{2} \)
$89$ \( ( -15 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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