Newform orbit 1352.2.f.f

• Label: 1352.2.f.f
• Level: $1352$
• Weight: $2$
• Character orbit: 1352.f
• Analytic conductor: $10.796$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7957743533\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.195105024.2
Defining polynomial:	\( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 104)
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.

\(f(q)\)	\(=\)	\( q + (\beta_1 - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{5} - \beta_{7} q^{7} + (\beta_{5} - \beta_1 + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} - 2\nu^{6} + 10\nu^{5} + 2\nu^{4} - 22\nu^{3} + 18\nu^{2} + 9\nu - 54 ) / 54 \)
\(\beta_{2}\)	\(=\)	\( ( -2\nu^{7} - \nu^{6} + 8\nu^{5} - 8\nu^{4} - 32\nu^{3} + 84\nu^{2} + 54\nu - 135 ) / 54 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{7} + 8\nu^{6} - 4\nu^{5} - 26\nu^{4} + 52\nu^{3} + 18\nu^{2} - 153\nu + 216 ) / 54 \)
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{7} + 14\nu^{6} - 10\nu^{5} - 14\nu^{4} + 58\nu^{3} - 30\nu^{2} - 63\nu + 216 ) / 54 \)
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{7} - 8\nu^{6} + 2\nu^{5} + 20\nu^{4} - 38\nu^{3} - 32\nu^{2} + 141\nu - 144 ) / 18 \)
\(\beta_{6}\)	\(=\)	\( ( 10\nu^{7} - 25\nu^{6} + 8\nu^{5} + 70\nu^{4} - 104\nu^{3} - 54\nu^{2} + 378\nu - 459 ) / 54 \)
\(\beta_{7}\)	\(=\)	\( ( -11\nu^{7} + 23\nu^{6} - 16\nu^{5} - 50\nu^{4} + 100\nu^{3} + 54\nu^{2} - 387\nu + 513 ) / 54 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(1015\)	\(1185\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

1.30512	−	1.13871i
1.30512	+	1.13871i
1.72124	−	0.193255i
1.72124	+	0.193255i
−1.58726	−	0.693255i
−1.58726	+	0.693255i
0.560908	+	1.63871i
0.560908	−	1.63871i

0

−3.27743

0

−

2.61023i

0

1.12182i

0

7.74153

0

337.2

0

−3.27743

0

2.61023i

0

−

1.12182i

0

7.74153

0

337.3

0

−1.38651

0

−

3.44247i

0

−

3.17452i

0

−1.07759

0

337.4

0

−1.38651

0

3.44247i

0

3.17452i

0

−1.07759

0

337.5

0

0.386509

0

−

3.17452i

0

−

3.44247i

0

−2.85061

0

337.6

0

0.386509

0

3.17452i

0

3.44247i

0

−2.85061

0

337.7

0

2.27743

0

−

1.12182i

0

2.61023i

0

2.18667

0

337.8

0

2.27743

0

1.12182i

0

−

2.61023i

0

2.18667

0

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	1352.2.f.f		8
4.b	odd	2	1		2704.2.f.q		8
13.b	even	2	1	inner	1352.2.f.f		8
13.c	even	3	1		104.2.o.a	✓	8
13.c	even	3	1		1352.2.o.f		8
13.d	odd	4	1		1352.2.a.k		4
13.d	odd	4	1		1352.2.a.l		4
13.e	even	6	1		104.2.o.a	✓	8
13.e	even	6	1		1352.2.o.f		8
13.f	odd	12	2		1352.2.i.k		8
13.f	odd	12	2		1352.2.i.l		8
39.h	odd	6	1		936.2.bi.b		8
39.i	odd	6	1		936.2.bi.b		8
52.b	odd	2	1		2704.2.f.q		8
52.f	even	4	1		2704.2.a.bd		4
52.f	even	4	1		2704.2.a.be		4
52.i	odd	6	1		208.2.w.c		8
52.j	odd	6	1		208.2.w.c		8
104.n	odd	6	1		832.2.w.i		8
104.p	odd	6	1		832.2.w.i		8
104.r	even	6	1		832.2.w.g		8
104.s	even	6	1		832.2.w.g		8
156.p	even	6	1		1872.2.by.n		8
156.r	even	6	1		1872.2.by.n		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.o.a	✓	8	13.c	even	3	1
104.2.o.a	✓	8	13.e	even	6	1
208.2.w.c		8	52.i	odd	6	1
208.2.w.c		8	52.j	odd	6	1
832.2.w.g		8	104.r	even	6	1
832.2.w.g		8	104.s	even	6	1
832.2.w.i		8	104.n	odd	6	1
832.2.w.i		8	104.p	odd	6	1
936.2.bi.b		8	39.h	odd	6	1
936.2.bi.b		8	39.i	odd	6	1
1352.2.a.k		4	13.d	odd	4	1
1352.2.a.l		4	13.d	odd	4	1
1352.2.f.f		8	1.a	even	1	1	trivial
1352.2.f.f		8	13.b	even	2	1	inner
1352.2.i.k		8	13.f	odd	12	2
1352.2.i.l		8	13.f	odd	12	2
1352.2.o.f		8	13.c	even	3	1
1352.2.o.f		8	13.e	even	6	1
1872.2.by.n		8	156.p	even	6	1
1872.2.by.n		8	156.r	even	6	1
2704.2.a.bd		4	52.f	even	4	1
2704.2.a.be		4	52.f	even	4	1
2704.2.f.q		8	4.b	odd	2	1
2704.2.f.q		8	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 8T_{3} + 4 \)

\( T_{5}^{8} + 30T_{5}^{6} + 305T_{5}^{4} + 1152T_{5}^{2} + 1024 \)

\( T_{11}^{8} + 30T_{11}^{6} + 305T_{11}^{4} + 1152T_{11}^{2} + 1024 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	(T^4 + 2*T^3 - 7*T^2 - 8*T + 4)^2
$5$	T^8 + 30*T^6 + 305*T^4 + 1152*T^2 + 1024
$7$	T^8 + 30*T^6 + 305*T^4 + 1152*T^2 + 1024
$11$	T^8 + 30*T^6 + 305*T^4 + 1152*T^2 + 1024