Newform orbit 546.2.k.d

• Label: 546.2.k.d
• 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:	8.0.6498455769.2
Defining polynomial:	\( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( 6\nu^{7} + 21\nu^{6} - 14\nu^{5} + 193\nu^{4} + 126\nu^{3} + 532\nu^{2} - 664\nu + 112 ) / 119 \)
\(\beta_{2}\)	\(=\)	\( ( 8\nu^{7} - 6\nu^{6} + 21\nu^{5} + 59\nu^{4} + 66\nu^{3} - 84\nu^{2} - 449\nu - 15 ) / 119 \)
\(\beta_{3}\)	\(=\)	\( ( -15\nu^{7} + 7\nu^{6} - 84\nu^{5} - 66\nu^{4} - 434\nu^{3} - 21\nu^{2} - 6\nu + 315 ) / 119 \)
\(\beta_{4}\)	\(=\)	\( ( -22\nu^{7} + 42\nu^{6} - 147\nu^{5} + 46\nu^{4} - 462\nu^{3} + 588\nu^{2} - 104\nu + 105 ) / 119 \)
\(\beta_{5}\)	\(=\)	\( ( -24\nu^{7} + 69\nu^{6} - 182\nu^{5} + 180\nu^{4} - 402\nu^{3} + 1085\nu^{2} - 81\nu - 6 ) / 119 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{7} - 3\nu^{6} + 14\nu^{5} - \nu^{4} + 54\nu^{3} - 28\nu^{2} + 47\nu - 4 ) / 7 \)
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{7} - 4\nu^{6} + 28\nu^{5} + 22\nu^{4} + 121\nu^{3} + 7\nu^{2} + 2\nu + 4 ) / 7 \)

Character values

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

\(n\)	\(157\)	\(365\)	\(379\)
\(\chi(n)\)	\(-\beta_{4}\)	\(1\)	\(-1 + \beta_{4}\)

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

0.271028	−	0.469434i
−0.922415	+	1.59767i
1.33821	−	2.31784i
−0.186817	+	0.323577i
0.271028	+	0.469434i
−0.922415	−	1.59767i
1.33821	+	2.31784i
−0.186817	−	0.323577i

0.500000

+

0.866025i

1.00000

−0.500000

+

0.866025i

−1.15139

+

1.99426i

0.500000

+

0.866025i

−0.964471

+

2.46370i

−1.00000

1.00000

−2.30278

373.2

0.500000

+

0.866025i

1.00000

−0.500000

+

0.866025i

−1.15139

+

1.99426i

0.500000

+

0.866025i

2.61586

+

0.396592i

−1.00000

1.00000

−2.30278

373.3

0.500000

+

0.866025i

1.00000

−0.500000

+

0.866025i

0.651388

−

1.12824i

0.500000

+

0.866025i

−2.36323

+

1.18960i

−1.00000

1.00000

1.30278

373.4

0.500000

+

0.866025i

1.00000

−0.500000

+

0.866025i

0.651388

−

1.12824i

0.500000

+

0.866025i

2.21184

−

1.45181i

−1.00000

1.00000

1.30278

445.1

0.500000

−

0.866025i

1.00000

−0.500000

−

0.866025i

−1.15139

−

1.99426i

0.500000

−

0.866025i

−0.964471

−

2.46370i

−1.00000

1.00000

−2.30278

445.2

0.500000

−

0.866025i

1.00000

−0.500000

−

0.866025i

−1.15139

−

1.99426i

0.500000

−

0.866025i

2.61586

−

0.396592i

−1.00000

1.00000

−2.30278

445.3

0.500000

−

0.866025i

1.00000

−0.500000

−

0.866025i

0.651388

+

1.12824i

0.500000

−

0.866025i

−2.36323

−

1.18960i

−1.00000

1.00000

1.30278

445.4

0.500000

−

0.866025i

1.00000

−0.500000

−

0.866025i

0.651388

+

1.12824i

0.500000

−

0.866025i

2.21184

+

1.45181i

−1.00000

1.00000

1.30278

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.d	yes	8
3.b	odd	2	1		1638.2.p.g		8
7.c	even	3	1		546.2.j.b	✓	8
13.c	even	3	1		546.2.j.b	✓	8
21.h	odd	6	1		1638.2.m.i		8
39.i	odd	6	1		1638.2.m.i		8
91.g	even	3	1	inner	546.2.k.d	yes	8
273.bm	odd	6	1		1638.2.p.g		8

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + T_{5}^{3} + 4T_{5}^{2} - 3T_{5} + 9 \) 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^4 + T^3 + 4*T^2 - 3*T + 9)^2
$7$	T^8 - 3*T^7 - 4*T^6 + 3*T^5 + 57*T^4 + 21*T^3 - 196*T^2 - 1029*T + 2401
$11$	(T^4 - 2*T^3 - 36*T^2 + 63*T + 27)^2