Newform orbit 1776.1.n.a

• Label: 1776.1.n.a
• Level: $1776$
• Weight: $1$
• Character orbit: 1776.n
• Self dual: yes
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -111, 37
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1776.n (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 111)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.5328.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} + 2 q^{7} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\)	\(223\)	\(593\)	\(1297\)	\(1333\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1553.1

0

0

−1.00000

0

0

0

2.00000

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
37.b	even	2	1	RM by \(\Q(\sqrt{37}) \)
111.d	odd	2	1	CM by \(\Q(\sqrt{-111}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1776.1.n.a		1
3.b	odd	2	1	CM	1776.1.n.a		1
4.b	odd	2	1		111.1.d.a	✓	1
12.b	even	2	1		111.1.d.a	✓	1
20.d	odd	2	1		2775.1.h.a		1
20.e	even	4	2		2775.1.b.a		2
36.f	odd	6	2		2997.1.n.b		2
36.h	even	6	2		2997.1.n.b		2
37.b	even	2	1	RM	1776.1.n.a		1
60.h	even	2	1		2775.1.h.a		1
60.l	odd	4	2		2775.1.b.a		2
111.d	odd	2	1	CM	1776.1.n.a		1
148.b	odd	2	1		111.1.d.a	✓	1
444.g	even	2	1		111.1.d.a	✓	1
740.g	odd	2	1		2775.1.h.a		1
740.m	even	4	2		2775.1.b.a		2
1332.z	even	6	2		2997.1.n.b		2
1332.bl	odd	6	2		2997.1.n.b		2
2220.p	even	2	1		2775.1.h.a		1
2220.bf	odd	4	2		2775.1.b.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
111.1.d.a	✓	1	4.b	odd	2	1
111.1.d.a	✓	1	12.b	even	2	1
111.1.d.a	✓	1	148.b	odd	2	1
111.1.d.a	✓	1	444.g	even	2	1
1776.1.n.a		1	1.a	even	1	1	trivial
1776.1.n.a		1	3.b	odd	2	1	CM
1776.1.n.a		1	37.b	even	2	1	RM
1776.1.n.a		1	111.d	odd	2	1	CM
2775.1.b.a		2	20.e	even	4	2
2775.1.b.a		2	60.l	odd	4	2
2775.1.b.a		2	740.m	even	4	2
2775.1.b.a		2	2220.bf	odd	4	2
2775.1.h.a		1	20.d	odd	2	1
2775.1.h.a		1	60.h	even	2	1
2775.1.h.a		1	740.g	odd	2	1
2775.1.h.a		1	2220.p	even	2	1
2997.1.n.b		2	36.f	odd	6	2
2997.1.n.b		2	36.h	even	6	2
2997.1.n.b		2	1332.z	even	6	2
2997.1.n.b		2	1332.bl	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1776, [\chi])\):

\( T_{5} \)

\( T_{7} - 2 \)

Hecke characteristic polynomials

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