Newform orbit 1444.1.h.b

• Label: 1444.1.h.b
• Level: $1444$
• Weight: $1$
• Character orbit: 1444.h
• Analytic conductor: $0.721$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $S_{4}$
• CM/RM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1444.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.720649878242\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.27436.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\)	\(723\)	\(1085\)
\(\chi(n)\)	\(1\)	\(\beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

69.1

−1.22474	+	0.707107i
1.22474	−	0.707107i
−1.22474	−	0.707107i
1.22474	+	0.707107i

0

−1.22474

+

0.707107i

0

−0.500000

−

0.866025i

0

1.00000

0

0.500000

−

0.866025i

0

69.2

0

1.22474

−

0.707107i

0

−0.500000

−

0.866025i

0

1.00000

0

0.500000

−

0.866025i

0

293.1

0

−1.22474

−

0.707107i

0

−0.500000

+

0.866025i

0

1.00000

0

0.500000

+

0.866025i

0

293.2

0

1.22474

+

0.707107i

0

−0.500000

+

0.866025i

0

1.00000

0

0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	inner
19.c	even	3	1	inner
19.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1444.1.h.b		4
19.b	odd	2	1	inner	1444.1.h.b		4
19.c	even	3	1		1444.1.c.a	✓	2
19.c	even	3	1	inner	1444.1.h.b		4
19.d	odd	6	1		1444.1.c.a	✓	2
19.d	odd	6	1	inner	1444.1.h.b		4
19.e	even	9	6		1444.1.j.b		12
19.f	odd	18	6		1444.1.j.b		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1444.1.c.a	✓	2	19.c	even	3	1
1444.1.c.a	✓	2	19.d	odd	6	1
1444.1.h.b		4	1.a	even	1	1	trivial
1444.1.h.b		4	19.b	odd	2	1	inner
1444.1.h.b		4	19.c	even	3	1	inner
1444.1.h.b		4	19.d	odd	6	1	inner
1444.1.j.b		12	19.e	even	9	6
1444.1.j.b		12	19.f	odd	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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