Embedded newform 1776.1.n.a.1553.1

• Label: 1776.1.n.a.1553.1
• Level: $1776$
• Weight: $1$
• Character: 1776.1553
• 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 \(\Q(\sqrt{-22 +6 \sqrt{-3}})\)
Stark unit:	Root of $x^{4} - 684x^{3} + 934x^{2} - 684x + 1$

Embedding invariants

Embedding label			1553.1
Character	\(\chi\)	\(=\)	1776.1553

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} +2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{21} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{37} +3.00000 q^{49} +2.00000 q^{63} +2.00000 q^{67} -2.00000 q^{73} +1.00000 q^{75} +1.00000 q^{81} +O(q^{100})\)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−1.00000	−1.00000
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	1.00000	1.00000
			\(41\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(67\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1776.1.n.a.1553.1		1
3.2	odd	2	CM	1776.1.n.a.1553.1		1
4.3	odd	2		111.1.d.a.110.1	✓	1
12.11	even	2		111.1.d.a.110.1	✓	1
20.3	even	4		2775.1.b.a.2774.2		2
20.7	even	4		2775.1.b.a.2774.1		2
20.19	odd	2		2775.1.h.a.776.1		1
36.7	odd	6		2997.1.n.b.1997.1		2
36.11	even	6		2997.1.n.b.1997.1		2
36.23	even	6		2997.1.n.b.998.1		2
36.31	odd	6		2997.1.n.b.998.1		2
37.36	even	2	RM	1776.1.n.a.1553.1		1
60.23	odd	4		2775.1.b.a.2774.2		2
60.47	odd	4		2775.1.b.a.2774.1		2
60.59	even	2		2775.1.h.a.776.1		1
111.110	odd	2	CM	1776.1.n.a.1553.1		1
148.147	odd	2		111.1.d.a.110.1	✓	1
444.443	even	2		111.1.d.a.110.1	✓	1
740.147	even	4		2775.1.b.a.2774.1		2
740.443	even	4		2775.1.b.a.2774.2		2
740.739	odd	2		2775.1.h.a.776.1		1
1332.295	odd	6		2997.1.n.b.1997.1		2
1332.443	even	6		2997.1.n.b.1997.1		2
1332.887	even	6		2997.1.n.b.998.1		2
1332.1183	odd	6		2997.1.n.b.998.1		2
2220.443	odd	4		2775.1.b.a.2774.2		2
2220.887	odd	4		2775.1.b.a.2774.1		2
2220.2219	even	2		2775.1.h.a.776.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.1.d.a.110.1	✓	1	4.3	odd	2
111.1.d.a.110.1	✓	1	12.11	even	2
111.1.d.a.110.1	✓	1	148.147	odd	2
111.1.d.a.110.1	✓	1	444.443	even	2
1776.1.n.a.1553.1		1	1.1	even	1	trivial
1776.1.n.a.1553.1		1	3.2	odd	2	CM
1776.1.n.a.1553.1		1	37.36	even	2	RM
1776.1.n.a.1553.1		1	111.110	odd	2	CM
2775.1.b.a.2774.1		2	20.7	even	4
2775.1.b.a.2774.1		2	60.47	odd	4
2775.1.b.a.2774.1		2	740.147	even	4
2775.1.b.a.2774.1		2	2220.887	odd	4
2775.1.b.a.2774.2		2	20.3	even	4
2775.1.b.a.2774.2		2	60.23	odd	4
2775.1.b.a.2774.2		2	740.443	even	4
2775.1.b.a.2774.2		2	2220.443	odd	4
2775.1.h.a.776.1		1	20.19	odd	2
2775.1.h.a.776.1		1	60.59	even	2
2775.1.h.a.776.1		1	740.739	odd	2
2775.1.h.a.776.1		1	2220.2219	even	2
2997.1.n.b.998.1		2	36.23	even	6
2997.1.n.b.998.1		2	36.31	odd	6
2997.1.n.b.998.1		2	1332.887	even	6
2997.1.n.b.998.1		2	1332.1183	odd	6
2997.1.n.b.1997.1		2	36.7	odd	6
2997.1.n.b.1997.1		2	36.11	even	6
2997.1.n.b.1997.1		2	1332.295	odd	6
2997.1.n.b.1997.1		2	1332.443	even	6