Newform orbit 845.2.c.c

• Label: 845.2.c.c
• Level: $845$
• Weight: $2$
• Character orbit: 845.c
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{5})\)
Defining polynomial:	\( x^{4} + 3x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (2 \beta_{2} + 1) q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + (2 \beta_{3} - \beta_1) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 1 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 2\nu \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

506.1

	−	1.61803i
	−	0.618034i
		0.618034i
		1.61803i

−

1.61803i

−2.23607

−0.618034

−

1.00000i

3.61803i

−

0.236068i

−

2.23607i

2.00000

−1.61803

506.2

−

0.618034i

2.23607

1.61803

1.00000i

−

1.38197i

−

4.23607i

−

2.23607i

2.00000

0.618034

506.3

0.618034i

2.23607

1.61803

−

1.00000i

1.38197i

4.23607i

2.23607i

2.00000

0.618034

506.4

1.61803i

−2.23607

−0.618034

1.00000i

−

3.61803i

0.236068i

2.23607i

2.00000

−1.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 3T^{2} + 1 \)
$3$	\( (T^{2} - 5)^{2} \)
$5$	\( (T^{2} + 1)^{2} \)
$7$	\( T^{4} + 18T^{2} + 1 \)
$11$	\( T^{4} + 18T^{2} + 1 \)