Newform orbit 1352.2.i.b

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7957743533\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(677\)	\(1015\)	\(1185\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

529.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−0.500000

+

0.866025i

0

−1.00000

0

−2.50000

−

4.33013i

0

1.00000

+

1.73205i

0

1329.1

0

−0.500000

−

0.866025i

0

−1.00000

0

−2.50000

+

4.33013i

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1352.2.i.b		2
13.b	even	2	1		1352.2.i.c		2
13.c	even	3	1		104.2.a.a	✓	1
13.c	even	3	1	inner	1352.2.i.b		2
13.d	odd	4	2		1352.2.o.a		4
13.e	even	6	1		1352.2.a.b		1
13.e	even	6	1		1352.2.i.c		2
13.f	odd	12	2		1352.2.f.b		2
13.f	odd	12	2		1352.2.o.a		4
39.i	odd	6	1		936.2.a.f		1
52.i	odd	6	1		2704.2.a.d		1
52.j	odd	6	1		208.2.a.b		1
52.l	even	12	2		2704.2.f.e		2
65.n	even	6	1		2600.2.a.e		1
65.q	odd	12	2		2600.2.d.f		2
91.n	odd	6	1		5096.2.a.c		1
104.n	odd	6	1		832.2.a.h		1
104.r	even	6	1		832.2.a.c		1
156.p	even	6	1		1872.2.a.l		1
208.bg	odd	12	2		3328.2.b.t		2
208.bj	even	12	2		3328.2.b.a		2
260.v	odd	6	1		5200.2.a.bb		1
312.bh	odd	6	1		7488.2.a.x		1
312.bn	even	6	1		7488.2.a.u		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.a	✓	1	13.c	even	3	1
208.2.a.b		1	52.j	odd	6	1
832.2.a.c		1	104.r	even	6	1
832.2.a.h		1	104.n	odd	6	1
936.2.a.f		1	39.i	odd	6	1
1352.2.a.b		1	13.e	even	6	1
1352.2.f.b		2	13.f	odd	12	2
1352.2.i.b		2	1.a	even	1	1	trivial
1352.2.i.b		2	13.c	even	3	1	inner
1352.2.i.c		2	13.b	even	2	1
1352.2.i.c		2	13.e	even	6	1
1352.2.o.a		4	13.d	odd	4	2
1352.2.o.a		4	13.f	odd	12	2
1872.2.a.l		1	156.p	even	6	1
2600.2.a.e		1	65.n	even	6	1
2600.2.d.f		2	65.q	odd	12	2
2704.2.a.d		1	52.i	odd	6	1
2704.2.f.e		2	52.l	even	12	2
3328.2.b.a		2	208.bj	even	12	2
3328.2.b.t		2	208.bg	odd	12	2
5096.2.a.c		1	91.n	odd	6	1
5200.2.a.bb		1	260.v	odd	6	1
7488.2.a.u		1	312.bn	even	6	1
7488.2.a.x		1	312.bh	odd	6	1

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \)

\( T_{5} + 1 \)

\( T_{7}^{2} + 5T_{7} + 25 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T + 1 \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} + 5T + 25 \)
$11$	\( T^{2} - 2T + 4 \)