Newform orbit 1600.2.a.z

• Label: 1600.2.a.z
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.a
• Self dual: yes
• Analytic conductor: $12.776$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -20
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(12.7760643234\)
Analytic rank:	\(0\)
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 160)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \beta - 1) q^{3} + ( - \beta + 3) q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \)

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

−3.23607

0

0

0

0.763932

0

7.47214

0

1.2

0

1.23607

0

0

0

5.23607

0

−1.47214

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.2.a.z		2
4.b	odd	2	1		1600.2.a.bd		2
5.b	even	2	1		1600.2.a.bd		2
5.c	odd	4	2		320.2.c.d		4
8.b	even	2	1		800.2.a.n		2
8.d	odd	2	1		800.2.a.j		2
15.e	even	4	2		2880.2.f.w		4
20.d	odd	2	1	CM	1600.2.a.z		2
20.e	even	4	2		320.2.c.d		4
24.f	even	2	1		7200.2.a.cb		2
24.h	odd	2	1		7200.2.a.cr		2
40.e	odd	2	1		800.2.a.n		2
40.f	even	2	1		800.2.a.j		2
40.i	odd	4	2		160.2.c.b	✓	4
40.k	even	4	2		160.2.c.b	✓	4
60.l	odd	4	2		2880.2.f.w		4
80.i	odd	4	2		1280.2.f.h		4
80.j	even	4	2		1280.2.f.h		4
80.s	even	4	2		1280.2.f.g		4
80.t	odd	4	2		1280.2.f.g		4
120.i	odd	2	1		7200.2.a.cb		2
120.m	even	2	1		7200.2.a.cr		2
120.q	odd	4	2		1440.2.f.i		4
120.w	even	4	2		1440.2.f.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.c.b	✓	4	40.i	odd	4	2
160.2.c.b	✓	4	40.k	even	4	2
320.2.c.d		4	5.c	odd	4	2
320.2.c.d		4	20.e	even	4	2
800.2.a.j		2	8.d	odd	2	1
800.2.a.j		2	40.f	even	2	1
800.2.a.n		2	8.b	even	2	1
800.2.a.n		2	40.e	odd	2	1
1280.2.f.g		4	80.s	even	4	2
1280.2.f.g		4	80.t	odd	4	2
1280.2.f.h		4	80.i	odd	4	2
1280.2.f.h		4	80.j	even	4	2
1440.2.f.i		4	120.q	odd	4	2
1440.2.f.i		4	120.w	even	4	2
1600.2.a.z		2	1.a	even	1	1	trivial
1600.2.a.z		2	20.d	odd	2	1	CM
1600.2.a.bd		2	4.b	odd	2	1
1600.2.a.bd		2	5.b	even	2	1
2880.2.f.w		4	15.e	even	4	2
2880.2.f.w		4	60.l	odd	4	2
7200.2.a.cb		2	24.f	even	2	1
7200.2.a.cb		2	120.i	odd	2	1
7200.2.a.cr		2	24.h	odd	2	1
7200.2.a.cr		2	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3}^{2} + 2T_{3} - 4 \)

\( T_{7}^{2} - 6T_{7} + 4 \)

\( T_{11} \)

\( T_{13} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2T - 4 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 6T + 4 \)
$11$	\( T^{2} \)