Newform orbit 845.2.b.f

• Label: 845.2.b.f
• Level: $845$
• Weight: $2$
• Character orbit: 845.b
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.49787136.1
Defining polynomial:	\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + \beta_{3} q^{3} + \beta_{4} q^{4} + ( - \beta_{7} - \beta_{2}) q^{5} + \beta_{7} q^{6} + ( - \beta_{6} - \beta_{2}) q^{7} + ( - \beta_{6} - \beta_{2}) q^{8} + 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 5\nu^{4} + 15\nu^{2} + 8 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 8 ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{6} + 5\nu^{4} + 15\nu^{2} + 44 ) / 20 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 4 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

339.1

−0.228425	+	1.39564i
0.228425	−	1.39564i
−1.09445	−	0.895644i
1.09445	+	0.895644i
1.09445	−	0.895644i
−1.09445	+	0.895644i
0.228425	+	1.39564i
−0.228425	−	1.39564i

−

2.18890i

−

1.00000i

−2.79129

2.18890

−

0.456850i

−2.18890

1.73205i

1.73205i

2.00000

−1.00000

−

4.79129i

339.2

−

2.18890i

1.00000i

−2.79129

−2.18890

−

0.456850i

2.18890

1.73205i

1.73205i

2.00000

−1.00000

+

4.79129i

339.3

−

0.456850i

−

1.00000i

1.79129

0.456850

−

2.18890i

−0.456850

−

1.73205i

−

1.73205i

2.00000

−1.00000

−

0.208712i

339.4

−

0.456850i

1.00000i

1.79129

−0.456850

−

2.18890i

0.456850

−

1.73205i

−

1.73205i

2.00000

−1.00000

+

0.208712i

339.5

0.456850i

−

1.00000i

1.79129

−0.456850

+

2.18890i

0.456850

1.73205i

1.73205i

2.00000

−1.00000

−

0.208712i

339.6

0.456850i

1.00000i

1.79129

0.456850

+

2.18890i

−0.456850

1.73205i

1.73205i

2.00000

−1.00000

+

0.208712i

339.7

2.18890i

−

1.00000i

−2.79129

−2.18890

+

0.456850i

2.18890

−

1.73205i

−

1.73205i

2.00000

−1.00000

−

4.79129i

339.8

2.18890i

1.00000i

−2.79129

2.18890

+

0.456850i

−2.18890

−

1.73205i

−

1.73205i

2.00000

−1.00000

+

4.79129i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
13.b	even	2	1	inner
65.d	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.b.f		8
5.b	even	2	1	inner	845.2.b.f		8
5.c	odd	4	1		4225.2.a.bj		4
5.c	odd	4	1		4225.2.a.bk		4
13.b	even	2	1	inner	845.2.b.f		8
13.c	even	3	1		845.2.n.c		8
13.c	even	3	1		845.2.n.d		8
13.d	odd	4	2		845.2.d.c		8
13.e	even	6	1		845.2.n.c		8
13.e	even	6	1		845.2.n.d		8
13.f	odd	12	2		65.2.l.a	✓	8
13.f	odd	12	2		845.2.l.c		8
39.k	even	12	2		585.2.bf.a		8
52.l	even	12	2		1040.2.df.b		8
65.d	even	2	1	inner	845.2.b.f		8
65.g	odd	4	2		845.2.d.c		8
65.h	odd	4	1		4225.2.a.bj		4
65.h	odd	4	1		4225.2.a.bk		4
65.l	even	6	1		845.2.n.c		8
65.l	even	6	1		845.2.n.d		8
65.n	even	6	1		845.2.n.c		8
65.n	even	6	1		845.2.n.d		8
65.o	even	12	1		325.2.n.b		4
65.o	even	12	1		325.2.n.c		4
65.s	odd	12	2		65.2.l.a	✓	8
65.s	odd	12	2		845.2.l.c		8
65.t	even	12	1		325.2.n.b		4
65.t	even	12	1		325.2.n.c		4
195.bh	even	12	2		585.2.bf.a		8
260.bc	even	12	2		1040.2.df.b		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.l.a	✓	8	13.f	odd	12	2
65.2.l.a	✓	8	65.s	odd	12	2
325.2.n.b		4	65.o	even	12	1
325.2.n.b		4	65.t	even	12	1
325.2.n.c		4	65.o	even	12	1
325.2.n.c		4	65.t	even	12	1
585.2.bf.a		8	39.k	even	12	2
585.2.bf.a		8	195.bh	even	12	2
845.2.b.f		8	1.a	even	1	1	trivial
845.2.b.f		8	5.b	even	2	1	inner
845.2.b.f		8	13.b	even	2	1	inner
845.2.b.f		8	65.d	even	2	1	inner
845.2.d.c		8	13.d	odd	4	2
845.2.d.c		8	65.g	odd	4	2
845.2.l.c		8	13.f	odd	12	2
845.2.l.c		8	65.s	odd	12	2
845.2.n.c		8	13.c	even	3	1
845.2.n.c		8	13.e	even	6	1
845.2.n.c		8	65.l	even	6	1
845.2.n.c		8	65.n	even	6	1
845.2.n.d		8	13.c	even	3	1
845.2.n.d		8	13.e	even	6	1
845.2.n.d		8	65.l	even	6	1
845.2.n.d		8	65.n	even	6	1
1040.2.df.b		8	52.l	even	12	2
1040.2.df.b		8	260.bc	even	12	2
4225.2.a.bj		4	5.c	odd	4	1
4225.2.a.bj		4	65.h	odd	4	1
4225.2.a.bk		4	5.c	odd	4	1
4225.2.a.bk		4	65.h	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{4} + 5T_{2}^{2} + 1 \)

\( T_{11}^{2} - 7 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + 5 T^{2} + 1)^{2} \)
$3$	\( (T^{2} + 1)^{4} \)
$5$	\( T^{8} - 34T^{4} + 625 \)
$7$	\( (T^{2} + 3)^{4} \)
$11$	\( (T^{2} - 7)^{4} \)