Newform orbit 845.2.a.e

• Label: 845.2.a.e
• Level: $845$
• Weight: $2$
• Character orbit: 845.a
• Self dual: yes
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + ( - 2 \beta + 1) q^{3} + (\beta - 1) q^{4} + q^{5} + ( - \beta - 2) q^{6} + ( - 2 \beta + 3) q^{7} + ( - 2 \beta + 1) q^{8} + 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 5 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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

−0.618034
1.61803

−0.618034

2.23607

−1.61803

1.00000

−1.38197

4.23607

2.23607

2.00000

−0.618034

1.2

1.61803

−2.23607

0.618034

1.00000

−3.61803

−0.236068

−2.23607

2.00000

1.61803

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(-1\)
\(13\)	\(1\)

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	845.2.a.e		2
3.b	odd	2	1		7605.2.a.ba		2
5.b	even	2	1		4225.2.a.u		2
13.b	even	2	1		845.2.a.b		2
13.c	even	3	2		65.2.e.a	✓	4
13.d	odd	4	2		845.2.c.c		4
13.e	even	6	2		845.2.e.g		4
13.f	odd	12	4		845.2.m.e		8
39.d	odd	2	1		7605.2.a.bf		2
39.i	odd	6	2		585.2.j.e		4
52.j	odd	6	2		1040.2.q.n		4
65.d	even	2	1		4225.2.a.y		2
65.n	even	6	2		325.2.e.b		4
65.q	odd	12	4		325.2.o.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.e.a	✓	4	13.c	even	3	2
325.2.e.b		4	65.n	even	6	2
325.2.o.a		8	65.q	odd	12	4
585.2.j.e		4	39.i	odd	6	2
845.2.a.b		2	13.b	even	2	1
845.2.a.e		2	1.a	even	1	1	trivial
845.2.c.c		4	13.d	odd	4	2
845.2.e.g		4	13.e	even	6	2
845.2.m.e		8	13.f	odd	12	4
1040.2.q.n		4	52.j	odd	6	2
4225.2.a.u		2	5.b	even	2	1
4225.2.a.y		2	65.d	even	2	1
7605.2.a.ba		2	3.b	odd	2	1
7605.2.a.bf		2	39.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(845))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 1 \)
$3$	\( T^{2} - 5 \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} - 4T - 1 \)
$11$	\( T^{2} - 4T - 1 \)