Newform orbit 6400.2.a.bi

• Label: 6400.2.a.bi
• Level: $6400$
• Weight: $2$
• Character orbit: 6400.a
• Self dual: yes
• Analytic conductor: $51.104$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -40
• Inner twists: $4$

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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 320)
Fricke sign:	\(+1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^7 - 3 * q^9 - 2 * q^11 - b * q^13 + 6 * q^19 - b * q^23 - b * q^37 - 2 * q^41 + 3*b * q^47 + 13 * q^49 - 3*b * q^53 - 14 * q^59 - 3*b * q^63 - 2*b * q^77 + 9 * q^81 - 14 * q^89 - 20 * q^91 + 6 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} - 4 q^{11} + 12 q^{19} - 4 q^{41} + 26 q^{49} - 28 q^{59} + 18 q^{81} - 28 q^{89} - 40 q^{91} + 12 q^{99}+O(q^{100}) \)

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

−0.618034
1.61803

0

0

0

0

0

−4.47214

0

−3.00000

0

1.2

0

0

0

0

0

4.47214

0

−3.00000

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by \(\Q(\sqrt{-10}) \)
5.b	even	2	1	inner
8.d	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.bi		2
4.b	odd	2	1		6400.2.a.bj		2
5.b	even	2	1	inner	6400.2.a.bi		2
5.c	odd	4	2		1280.2.c.b		2
8.b	even	2	1		6400.2.a.bj		2
8.d	odd	2	1	inner	6400.2.a.bi		2
16.e	even	4	2		1600.2.d.g		4
16.f	odd	4	2		1600.2.d.g		4
20.d	odd	2	1		6400.2.a.bj		2
20.e	even	4	2		1280.2.c.c		2
40.e	odd	2	1	CM	6400.2.a.bi		2
40.f	even	2	1		6400.2.a.bj		2
40.i	odd	4	2		1280.2.c.c		2
40.k	even	4	2		1280.2.c.b		2
80.i	odd	4	2		320.2.f.a	✓	4
80.j	even	4	2		320.2.f.a	✓	4
80.k	odd	4	2		1600.2.d.g		4
80.q	even	4	2		1600.2.d.g		4
80.s	even	4	2		320.2.f.a	✓	4
80.t	odd	4	2		320.2.f.a	✓	4
240.z	odd	4	2		2880.2.d.e		4
240.bb	even	4	2		2880.2.d.e		4
240.bd	odd	4	2		2880.2.d.e		4
240.bf	even	4	2		2880.2.d.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
320.2.f.a	✓	4	80.i	odd	4	2
320.2.f.a	✓	4	80.j	even	4	2
320.2.f.a	✓	4	80.s	even	4	2
320.2.f.a	✓	4	80.t	odd	4	2
1280.2.c.b		2	5.c	odd	4	2
1280.2.c.b		2	40.k	even	4	2
1280.2.c.c		2	20.e	even	4	2
1280.2.c.c		2	40.i	odd	4	2
1600.2.d.g		4	16.e	even	4	2
1600.2.d.g		4	16.f	odd	4	2
1600.2.d.g		4	80.k	odd	4	2
1600.2.d.g		4	80.q	even	4	2
2880.2.d.e		4	240.z	odd	4	2
2880.2.d.e		4	240.bb	even	4	2
2880.2.d.e		4	240.bd	odd	4	2
2880.2.d.e		4	240.bf	even	4	2
6400.2.a.bi		2	1.a	even	1	1	trivial
6400.2.a.bi		2	5.b	even	2	1	inner
6400.2.a.bi		2	8.d	odd	2	1	inner
6400.2.a.bi		2	40.e	odd	2	1	CM
6400.2.a.bj		2	4.b	odd	2	1
6400.2.a.bj		2	8.b	even	2	1
6400.2.a.bj		2	20.d	odd	2	1
6400.2.a.bj		2	40.f	even	2	1

Hecke kernels

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

\( T_{3} \)

\( T_{7}^{2} - 20 \)

\( T_{11} + 2 \)

\( T_{13}^{2} - 20 \)

\( T_{17} \)

\( T_{29} \)

\( T_{31} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 20 \)
$11$	\( (T + 2)^{2} \)