Embedded newform 76.1.c.a.37.1

• Label: 76.1.c.a.37.1
• Level: $76$
• Weight: $1$
• Character: 76.37
• Self dual: yes
• Analytic conductor: $0.038$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.0379289409601\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.76.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.76.1
Stark unit:	Root of $x^{3} - 3x^{2} + x - 1$

Embedding invariants

Embedding label			37.1
Character	\(\chi\)	\(=\)	76.37

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(-1\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(5\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(7\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(11\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(19\)	1.00000	1.00000
			\(23\)	2.00000	2.00000	1.00000			\(0\)
1.00000						\(0\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(47\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.1.c.a.37.1	✓	1
3.2	odd	2		684.1.h.a.37.1		1
4.3	odd	2		304.1.e.a.113.1		1
5.2	odd	4		1900.1.g.a.949.1		2
5.3	odd	4		1900.1.g.a.949.2		2
5.4	even	2		1900.1.e.a.1101.1		1
7.2	even	3		3724.1.bc.c.949.1		2
7.3	odd	6		3724.1.bc.b.569.1		2
7.4	even	3		3724.1.bc.c.569.1		2
7.5	odd	6		3724.1.bc.b.949.1		2
7.6	odd	2		3724.1.e.c.1177.1		1
8.3	odd	2		1216.1.e.b.1025.1		1
8.5	even	2		1216.1.e.a.1025.1		1
12.11	even	2		2736.1.o.b.721.1		1
19.2	odd	18		1444.1.j.a.1345.1		6
19.3	odd	18		1444.1.j.a.333.1		6
19.4	even	9		1444.1.j.a.1029.1		6
19.5	even	9		1444.1.j.a.849.1		6
19.6	even	9		1444.1.j.a.477.1		6
19.7	even	3		1444.1.h.a.293.1		2
19.8	odd	6		1444.1.h.a.69.1		2
19.9	even	9		1444.1.j.a.1021.1		6
19.10	odd	18		1444.1.j.a.1021.1		6
19.11	even	3		1444.1.h.a.69.1		2
19.12	odd	6		1444.1.h.a.293.1		2
19.13	odd	18		1444.1.j.a.477.1		6
19.14	odd	18		1444.1.j.a.849.1		6
19.15	odd	18		1444.1.j.a.1029.1		6
19.16	even	9		1444.1.j.a.333.1		6
19.17	even	9		1444.1.j.a.1345.1		6
19.18	odd	2	CM	76.1.c.a.37.1	✓	1
57.56	even	2		684.1.h.a.37.1		1
76.75	even	2		304.1.e.a.113.1		1
95.18	even	4		1900.1.g.a.949.2		2
95.37	even	4		1900.1.g.a.949.1		2
95.94	odd	2		1900.1.e.a.1101.1		1
133.18	odd	6		3724.1.bc.c.569.1		2
133.37	odd	6		3724.1.bc.c.949.1		2
133.75	even	6		3724.1.bc.b.949.1		2
133.94	even	6		3724.1.bc.b.569.1		2
133.132	even	2		3724.1.e.c.1177.1		1
152.37	odd	2		1216.1.e.a.1025.1		1
152.75	even	2		1216.1.e.b.1025.1		1
228.227	odd	2		2736.1.o.b.721.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.1.c.a.37.1	✓	1	1.1	even	1	trivial
76.1.c.a.37.1	✓	1	19.18	odd	2	CM
304.1.e.a.113.1		1	4.3	odd	2
304.1.e.a.113.1		1	76.75	even	2
684.1.h.a.37.1		1	3.2	odd	2
684.1.h.a.37.1		1	57.56	even	2
1216.1.e.a.1025.1		1	8.5	even	2
1216.1.e.a.1025.1		1	152.37	odd	2
1216.1.e.b.1025.1		1	8.3	odd	2
1216.1.e.b.1025.1		1	152.75	even	2
1444.1.h.a.69.1		2	19.8	odd	6
1444.1.h.a.69.1		2	19.11	even	3
1444.1.h.a.293.1		2	19.7	even	3
1444.1.h.a.293.1		2	19.12	odd	6
1444.1.j.a.333.1		6	19.3	odd	18
1444.1.j.a.333.1		6	19.16	even	9
1444.1.j.a.477.1		6	19.6	even	9
1444.1.j.a.477.1		6	19.13	odd	18
1444.1.j.a.849.1		6	19.5	even	9
1444.1.j.a.849.1		6	19.14	odd	18
1444.1.j.a.1021.1		6	19.9	even	9
1444.1.j.a.1021.1		6	19.10	odd	18
1444.1.j.a.1029.1		6	19.4	even	9
1444.1.j.a.1029.1		6	19.15	odd	18
1444.1.j.a.1345.1		6	19.2	odd	18
1444.1.j.a.1345.1		6	19.17	even	9
1900.1.e.a.1101.1		1	5.4	even	2
1900.1.e.a.1101.1		1	95.94	odd	2
1900.1.g.a.949.1		2	5.2	odd	4
1900.1.g.a.949.1		2	95.37	even	4
1900.1.g.a.949.2		2	5.3	odd	4
1900.1.g.a.949.2		2	95.18	even	4
2736.1.o.b.721.1		1	12.11	even	2
2736.1.o.b.721.1		1	228.227	odd	2
3724.1.e.c.1177.1		1	7.6	odd	2
3724.1.e.c.1177.1		1	133.132	even	2
3724.1.bc.b.569.1		2	7.3	odd	6
3724.1.bc.b.569.1		2	133.94	even	6
3724.1.bc.b.949.1		2	7.5	odd	6
3724.1.bc.b.949.1		2	133.75	even	6
3724.1.bc.c.569.1		2	7.4	even	3
3724.1.bc.c.569.1		2	133.18	odd	6
3724.1.bc.c.949.1		2	7.2	even	3
3724.1.bc.c.949.1		2	133.37	odd	6