Newform orbit 1444.1.b.a

• Label: 1444.1.b.a
• Level: $1444$
• Weight: $1$
• Character orbit: 1444.b
• Self dual: yes
• Analytic conductor: $0.721$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1444.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.720649878242\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.2085136.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + q^{4} + (\beta - 1) q^{5} - q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(723\)	\(1085\)
\(\chi(n)\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

723.1

−0.618034
1.61803

−1.00000

0

1.00000

−1.61803

0

0

−1.00000

1.00000

1.61803

723.2

−1.00000

0

1.00000

0.618034

0

0

−1.00000

1.00000

−0.618034

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1444.1.b.a	✓	2
4.b	odd	2	1	CM	1444.1.b.a	✓	2
19.b	odd	2	1		1444.1.b.b	yes	2
19.c	even	3	2		1444.1.g.b		4
19.d	odd	6	2		1444.1.g.a		4
19.e	even	9	6		1444.1.l.b		12
19.f	odd	18	6		1444.1.l.a		12
76.d	even	2	1		1444.1.b.b	yes	2
76.f	even	6	2		1444.1.g.a		4
76.g	odd	6	2		1444.1.g.b		4
76.k	even	18	6		1444.1.l.a		12
76.l	odd	18	6		1444.1.l.b		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1444.1.b.a	✓	2	1.a	even	1	1	trivial
1444.1.b.a	✓	2	4.b	odd	2	1	CM
1444.1.b.b	yes	2	19.b	odd	2	1
1444.1.b.b	yes	2	76.d	even	2	1
1444.1.g.a		4	19.d	odd	6	2
1444.1.g.a		4	76.f	even	6	2
1444.1.g.b		4	19.c	even	3	2
1444.1.g.b		4	76.g	odd	6	2
1444.1.l.a		12	19.f	odd	18	6
1444.1.l.a		12	76.k	even	18	6
1444.1.l.b		12	19.e	even	9	6
1444.1.l.b		12	76.l	odd	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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