Newform orbit 468.1.y.a

• Label: 468.1.y.a
• Level: $468$
• Weight: $1$
• Character orbit: 468.y
• Analytic conductor: $0.234$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $A_{4}$
• CM/RM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	468.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.233562425912\)
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 4.0.219024.1
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(209\)	\(235\)
\(\chi(n)\)	\(-\zeta_{12}^{2}\)	\(\zeta_{12}^{4}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

139.1

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

−

1.00000i

1.00000i

−1.00000

0.500000

−

0.866025i

1.00000

0.866025

+

0.500000i

1.00000i

−1.00000

−0.866025

−

0.500000i

139.2

1.00000i

−

1.00000i

−1.00000

0.500000

−

0.866025i

1.00000

−0.866025

−

0.500000i

−

1.00000i

−1.00000

0.866025

+

0.500000i

367.1

−

1.00000i

1.00000i

−1.00000

0.500000

+

0.866025i

1.00000

−0.866025

+

0.500000i

1.00000i

−1.00000

0.866025

−

0.500000i

367.2

1.00000i

−

1.00000i

−1.00000

0.500000

+

0.866025i

1.00000

0.866025

−

0.500000i

−

1.00000i

−1.00000

−0.866025

+

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
117.h	even	3	1	inner
468.y	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.1.y.a	yes	4
3.b	odd	2	1		1404.1.y.a		4
4.b	odd	2	1	inner	468.1.y.a	yes	4
9.c	even	3	1		468.1.v.a	✓	4
9.d	odd	6	1		1404.1.v.a		4
12.b	even	2	1		1404.1.y.a		4
13.c	even	3	1		468.1.v.a	✓	4
36.f	odd	6	1		468.1.v.a	✓	4
36.h	even	6	1		1404.1.v.a		4
39.i	odd	6	1		1404.1.v.a		4
52.j	odd	6	1		468.1.v.a	✓	4
117.h	even	3	1	inner	468.1.y.a	yes	4
117.k	odd	6	1		1404.1.y.a		4
156.p	even	6	1		1404.1.v.a		4
468.y	odd	6	1	inner	468.1.y.a	yes	4
468.bm	even	6	1		1404.1.y.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.1.v.a	✓	4	9.c	even	3	1
468.1.v.a	✓	4	13.c	even	3	1
468.1.v.a	✓	4	36.f	odd	6	1
468.1.v.a	✓	4	52.j	odd	6	1
468.1.y.a	yes	4	1.a	even	1	1	trivial
468.1.y.a	yes	4	4.b	odd	2	1	inner
468.1.y.a	yes	4	117.h	even	3	1	inner
468.1.y.a	yes	4	468.y	odd	6	1	inner
1404.1.v.a		4	9.d	odd	6	1
1404.1.v.a		4	36.h	even	6	1
1404.1.v.a		4	39.i	odd	6	1
1404.1.v.a		4	156.p	even	6	1
1404.1.y.a		4	3.b	odd	2	1
1404.1.y.a		4	12.b	even	2	1
1404.1.y.a		4	117.k	odd	6	1
1404.1.y.a		4	468.bm	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

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