Newform orbit 1352.2.o.d

• Label: 1352.2.o.d
• Level: $1352$
• Weight: $2$
• Character orbit: 1352.o
• Analytic conductor: $10.796$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7957743533\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.1731891456.1
Defining polynomial:	\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) :

\(\beta_{1}\)	\(=\)	\( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} - 441\nu ) / 260 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{6} + 116 ) / 65 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - 181\nu ) / 130 \)
\(\beta_{5}\)	\(=\)	\( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 9\nu^{5} + 65\nu^{3} - 144\nu ) / 64 \)
\(\beta_{7}\)	\(=\)	\( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 416 \)

Character values

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

\(n\)	\(677\)	\(1015\)	\(1185\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

−1.35234	−	0.780776i
1.35234	+	0.780776i
−2.21837	−	1.28078i
2.21837	+	1.28078i
1.35234	−	0.780776i
−1.35234	+	0.780776i
2.21837	−	1.28078i
−2.21837	+	1.28078i

0

−1.28078

−

2.21837i

0

−

0.561553i

0

−2.21837

−

1.28078i

0

−1.78078

+

3.08440i

0

361.2

0

−1.28078

−

2.21837i

0

0.561553i

0

2.21837

+

1.28078i

0

−1.78078

+

3.08440i

0

361.3

0

0.780776

+

1.35234i

0

−

3.56155i

0

−1.35234

−

0.780776i

0

0.280776

−

0.486319i

0

361.4

0

0.780776

+

1.35234i

0

3.56155i

0

1.35234

+

0.780776i

0

0.280776

−

0.486319i

0

1161.1

0

−1.28078

+

2.21837i

0

−

0.561553i

0

2.21837

−

1.28078i

0

−1.78078

−

3.08440i

0

1161.2

0

−1.28078

+

2.21837i

0

0.561553i

0

−2.21837

+

1.28078i

0

−1.78078

−

3.08440i

0

1161.3

0

0.780776

−

1.35234i

0

−

3.56155i

0

1.35234

−

0.780776i

0

0.280776

+

0.486319i

0

1161.4

0

0.780776

−

1.35234i

0

3.56155i

0

−1.35234

+

0.780776i

0

0.280776

+

0.486319i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1352.2.o.d		8
13.b	even	2	1	inner	1352.2.o.d		8
13.c	even	3	1		1352.2.f.c		4
13.c	even	3	1	inner	1352.2.o.d		8
13.d	odd	4	1		1352.2.i.d		4
13.d	odd	4	1		1352.2.i.f		4
13.e	even	6	1		1352.2.f.c		4
13.e	even	6	1	inner	1352.2.o.d		8
13.f	odd	12	1		104.2.a.b	✓	2
13.f	odd	12	1		1352.2.a.g		2
13.f	odd	12	1		1352.2.i.d		4
13.f	odd	12	1		1352.2.i.f		4
39.k	even	12	1		936.2.a.j		2
52.i	odd	6	1		2704.2.f.k		4
52.j	odd	6	1		2704.2.f.k		4
52.l	even	12	1		208.2.a.e		2
52.l	even	12	1		2704.2.a.p		2
65.o	even	12	1		2600.2.d.k		4
65.s	odd	12	1		2600.2.a.p		2
65.t	even	12	1		2600.2.d.k		4
91.bc	even	12	1		5096.2.a.m		2
104.u	even	12	1		832.2.a.n		2
104.x	odd	12	1		832.2.a.k		2
156.v	odd	12	1		1872.2.a.u		2
208.be	odd	12	1		3328.2.b.y		4
208.bf	even	12	1		3328.2.b.w		4
208.bk	even	12	1		3328.2.b.w		4
208.bl	odd	12	1		3328.2.b.y		4
260.bc	even	12	1		5200.2.a.bw		2
312.bo	even	12	1		7488.2.a.cu		2
312.bq	odd	12	1		7488.2.a.cv		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.b	✓	2	13.f	odd	12	1
208.2.a.e		2	52.l	even	12	1
832.2.a.k		2	104.x	odd	12	1
832.2.a.n		2	104.u	even	12	1
936.2.a.j		2	39.k	even	12	1
1352.2.a.g		2	13.f	odd	12	1
1352.2.f.c		4	13.c	even	3	1
1352.2.f.c		4	13.e	even	6	1
1352.2.i.d		4	13.d	odd	4	1
1352.2.i.d		4	13.f	odd	12	1
1352.2.i.f		4	13.d	odd	4	1
1352.2.i.f		4	13.f	odd	12	1
1352.2.o.d		8	1.a	even	1	1	trivial
1352.2.o.d		8	13.b	even	2	1	inner
1352.2.o.d		8	13.c	even	3	1	inner
1352.2.o.d		8	13.e	even	6	1	inner
1872.2.a.u		2	156.v	odd	12	1
2600.2.a.p		2	65.s	odd	12	1
2600.2.d.k		4	65.o	even	12	1
2600.2.d.k		4	65.t	even	12	1
2704.2.a.p		2	52.l	even	12	1
2704.2.f.k		4	52.i	odd	6	1
2704.2.f.k		4	52.j	odd	6	1
3328.2.b.w		4	208.bf	even	12	1
3328.2.b.w		4	208.bk	even	12	1
3328.2.b.y		4	208.be	odd	12	1
3328.2.b.y		4	208.bl	odd	12	1
5096.2.a.m		2	91.bc	even	12	1
5200.2.a.bw		2	260.bc	even	12	1
7488.2.a.cu		2	312.bo	even	12	1
7488.2.a.cv		2	312.bq	odd	12	1

Hecke kernels

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

\( T_{3}^{4} + T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 16 \)

\( T_{5}^{4} + 13T_{5}^{2} + 4 \)

\( T_{11}^{8} - 36T_{11}^{6} + 1040T_{11}^{4} - 9216T_{11}^{2} + 65536 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	(T^4 + T^3 + 5*T^2 - 4*T + 16)^2
$5$	\( (T^{4} + 13 T^{2} + 4)^{2} \)
$7$	T^8 - 9*T^6 + 65*T^4 - 144*T^2 + 256
$11$	T^8 - 36*T^6 + 1040*T^4 - 9216*T^2 + 65536