Newform orbit 546.2.j.a

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.35983195036\)
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:	yes
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.

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

Character values

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

\(n\)	\(157\)	\(365\)	\(379\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

289.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.00000

0.500000

−

0.866025i

1.00000

0

0.500000

−

0.866025i

0.500000

+

2.59808i

1.00000

−0.500000

−

0.866025i

0

529.1

1.00000

0.500000

+

0.866025i

1.00000

0

0.500000

+

0.866025i

0.500000

−

2.59808i

1.00000

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.h	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.j.a	✓	2
3.b	odd	2	1		1638.2.m.a		2
7.c	even	3	1		546.2.k.a	yes	2
13.c	even	3	1		546.2.k.a	yes	2
21.h	odd	6	1		1638.2.p.d		2
39.i	odd	6	1		1638.2.p.d		2
91.h	even	3	1	inner	546.2.j.a	✓	2
273.s	odd	6	1		1638.2.m.a		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

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