Newform orbit 546.2.k.c

• Label: 546.2.k.c
• Level: $546$
• Weight: $2$
• Character orbit: 546.k
• Analytic conductor: $4.360$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.35983195036\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.447703281.1
Defining polynomial:	\( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 - b2 * q^2 + q^3 + (b2 - 1) * q^4 + (-b7 + b4 - b3 - b2 + 1) * q^5 - b2 * q^6 + (-b7 + b6 - b5 - b3) * q^7 + q^8 + q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 8 q^{3} - 4 q^{4} + 2 q^{5} - 4 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} + 4\nu^{5} + 2\nu^{4} + \nu^{3} - 10\nu^{2} + 16 ) / 8 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 8 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{7} - \nu^{6} + 2\nu^{5} + 2\nu^{4} - \nu^{3} - 16\nu^{2} - 8\nu + 8 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 16\nu^{2} + 4\nu - 24 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} - \nu^{6} - 4\nu^{5} + 3\nu^{3} + 8\nu^{2} - 2\nu - 20 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 24\nu^{2} + 12\nu - 32 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( 4\nu^{7} + \nu^{6} - 9\nu^{5} - 6\nu^{4} + 6\nu^{3} + 27\nu^{2} + 4\nu - 40 ) / 4 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\)	\(157\)	\(365\)	\(379\)
\(\chi(n)\)	\(-1 + \beta_{2}\)	\(1\)	\(-\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

373.1

1.19003	−	0.764088i
−0.571299	+	1.29368i
1.26359	+	0.635098i
−1.38232	−	0.298668i
1.19003	+	0.764088i
−0.571299	−	1.29368i
1.26359	−	0.635098i
−1.38232	+	0.298668i

−0.500000

−

0.866025i

1.00000

−0.500000

+

0.866025i

−0.924396

+

1.60110i

−0.500000

−

0.866025i

−1.65876

+

2.06119i

1.00000

1.00000

1.84879

373.2

−0.500000

−

0.866025i

1.00000

−0.500000

+

0.866025i

−0.441221

+

0.764218i

−0.500000

−

0.866025i

−2.45374

−

0.989520i

1.00000

1.00000

0.882443

373.3

−0.500000

−

0.866025i

1.00000

−0.500000

+

0.866025i

0.611519

−

1.05918i

−0.500000

−

0.866025i

1.48662

+

2.18860i

1.00000

1.00000

−1.22304

373.4

−0.500000

−

0.866025i

1.00000

−0.500000

+

0.866025i

1.75410

−

3.03819i

−0.500000

−

0.866025i

1.12588

−

2.39424i

1.00000

1.00000

−3.50820

445.1

−0.500000

+

0.866025i

1.00000

−0.500000

−

0.866025i

−0.924396

−

1.60110i

−0.500000

+

0.866025i

−1.65876

−

2.06119i

1.00000

1.00000

1.84879

445.2

−0.500000

+

0.866025i

1.00000

−0.500000

−

0.866025i

−0.441221

−

0.764218i

−0.500000

+

0.866025i

−2.45374

+

0.989520i

1.00000

1.00000

0.882443

445.3

−0.500000

+

0.866025i

1.00000

−0.500000

−

0.866025i

0.611519

+

1.05918i

−0.500000

+

0.866025i

1.48662

−

2.18860i

1.00000

1.00000

−1.22304

445.4

−0.500000

+

0.866025i

1.00000

−0.500000

−

0.866025i

1.75410

+

3.03819i

−0.500000

+

0.866025i

1.12588

+

2.39424i

1.00000

1.00000

−3.50820

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	546.2.k.c	yes	8
3.b	odd	2	1		1638.2.p.h		8
7.c	even	3	1		546.2.j.c	✓	8
13.c	even	3	1		546.2.j.c	✓	8
21.h	odd	6	1		1638.2.m.h		8
39.i	odd	6	1		1638.2.m.h		8
91.g	even	3	1	inner	546.2.k.c	yes	8
273.bm	odd	6	1		1638.2.p.h		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.2.j.c	✓	8	7.c	even	3	1
546.2.j.c	✓	8	13.c	even	3	1
546.2.k.c	yes	8	1.a	even	1	1	trivial
546.2.k.c	yes	8	91.g	even	3	1	inner
1638.2.m.h		8	21.h	odd	6	1
1638.2.m.h		8	39.i	odd	6	1
1638.2.p.h		8	3.b	odd	2	1
1638.2.p.h		8	273.bm	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{7} + 11T_{5}^{6} + 6T_{5}^{5} + 50T_{5}^{4} + 65T_{5}^{2} + 28T_{5} + 49 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{4} \)
$3$	\( (T - 1)^{8} \)
$5$	T^8 - 2*T^7 + 11*T^6 + 6*T^5 + 50*T^4 + 65*T^2 + 28*T + 49
$7$	T^8 + 3*T^7 + 8*T^6 + 33*T^5 + 123*T^4 + 231*T^3 + 392*T^2 + 1029*T + 2401
$11$	(T^4 + 4*T^3 - 14*T^2 + 9*T - 1)^2