Newform orbit 684.1.x.a

• Label: 684.1.x.a
• Level: $684$
• Weight: $1$
• Character orbit: 684.x
• Analytic conductor: $0.341$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $A_{4}$
• CM/RM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	684.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.341360468641\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(A_{4}\)
Projective field:	Galois closure of 4.0.467856.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - q^{2} - \zeta_{12}^{3} q^{3} + q^{4} + \zeta_{12}^{2} q^{5} + \zeta_{12}^{3} q^{6} - \zeta_{12}^{5} q^{7} - q^{8} - q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(343\)	\(533\)
\(\chi(n)\)	\(-\zeta_{12}^{2}\)	\(-1\)	\(\zeta_{12}^{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

−0.866025	+	0.500000i
0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	−	0.500000i

−1.00000

−

1.00000i

1.00000

0.500000

−

0.866025i

1.00000i

−0.866025

−

0.500000i

−1.00000

−1.00000

−0.500000

+

0.866025i

7.2

−1.00000

1.00000i

1.00000

0.500000

−

0.866025i

−

1.00000i

0.866025

+

0.500000i

−1.00000

−1.00000

−0.500000

+

0.866025i

391.1

−1.00000

−

1.00000i

1.00000

0.500000

+

0.866025i

1.00000i

0.866025

−

0.500000i

−1.00000

−1.00000

−0.500000

−

0.866025i

391.2

−1.00000

1.00000i

1.00000

0.500000

+

0.866025i

−

1.00000i

−0.866025

+

0.500000i

−1.00000

−1.00000

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.h	even	3	1	inner
684.x	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.1.x.a	✓	4
3.b	odd	2	1		2052.1.x.a		4
4.b	odd	2	1	inner	684.1.x.a	✓	4
9.c	even	3	1		684.1.bm.a	yes	4
9.d	odd	6	1		2052.1.bm.a		4
12.b	even	2	1		2052.1.x.a		4
19.c	even	3	1		684.1.bm.a	yes	4
36.f	odd	6	1		684.1.bm.a	yes	4
36.h	even	6	1		2052.1.bm.a		4
57.h	odd	6	1		2052.1.bm.a		4
76.g	odd	6	1		684.1.bm.a	yes	4
171.h	even	3	1	inner	684.1.x.a	✓	4
171.j	odd	6	1		2052.1.x.a		4
228.m	even	6	1		2052.1.bm.a		4
684.x	odd	6	1	inner	684.1.x.a	✓	4
684.bi	even	6	1		2052.1.x.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.1.x.a	✓	4	1.a	even	1	1	trivial
684.1.x.a	✓	4	4.b	odd	2	1	inner
684.1.x.a	✓	4	171.h	even	3	1	inner
684.1.x.a	✓	4	684.x	odd	6	1	inner
684.1.bm.a	yes	4	9.c	even	3	1
684.1.bm.a	yes	4	19.c	even	3	1
684.1.bm.a	yes	4	36.f	odd	6	1
684.1.bm.a	yes	4	76.g	odd	6	1
2052.1.x.a		4	3.b	odd	2	1
2052.1.x.a		4	12.b	even	2	1
2052.1.x.a		4	171.j	odd	6	1
2052.1.x.a		4	684.bi	even	6	1
2052.1.bm.a		4	9.d	odd	6	1
2052.1.bm.a		4	36.h	even	6	1
2052.1.bm.a		4	57.h	odd	6	1
2052.1.bm.a		4	228.m	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

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