Embedded newform 304.1.e.a.113.1

• Label: 304.1.e.a.113.1
• Level: $304$
• Weight: $1$
• Character: 304.113
• Self dual: yes
• Analytic conductor: $0.152$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.151715763840\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 76)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.76.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.92416.1
Stark unit:	Root of $x^{6} - 10x^{5} - 41x^{4} - 204x^{3} - 41x^{2} - 10x + 1$

Embedding invariants

Embedding label			113.1
Character	\(\chi\)	\(=\)	304.113

$q$-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 \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

Coefficient data

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

(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	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(11\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(13\)	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			\(\pi\)
−1.00000						\(\pi\)
\(29\)	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	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(47\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.1.e.a.113.1		1
3.2	odd	2		2736.1.o.b.721.1		1
4.3	odd	2		76.1.c.a.37.1	✓	1
8.3	odd	2		1216.1.e.a.1025.1		1
8.5	even	2		1216.1.e.b.1025.1		1
12.11	even	2		684.1.h.a.37.1		1
19.18	odd	2	CM	304.1.e.a.113.1		1
20.3	even	4		1900.1.g.a.949.2		2
20.7	even	4		1900.1.g.a.949.1		2
20.19	odd	2		1900.1.e.a.1101.1		1
28.3	even	6		3724.1.bc.b.569.1		2
28.11	odd	6		3724.1.bc.c.569.1		2
28.19	even	6		3724.1.bc.b.949.1		2
28.23	odd	6		3724.1.bc.c.949.1		2
28.27	even	2		3724.1.e.c.1177.1		1
57.56	even	2		2736.1.o.b.721.1		1
76.3	even	18		1444.1.j.a.333.1		6
76.7	odd	6		1444.1.h.a.293.1		2
76.11	odd	6		1444.1.h.a.69.1		2
76.15	even	18		1444.1.j.a.1029.1		6
76.23	odd	18		1444.1.j.a.1029.1		6
76.27	even	6		1444.1.h.a.69.1		2
76.31	even	6		1444.1.h.a.293.1		2
76.35	odd	18		1444.1.j.a.333.1		6
76.43	odd	18		1444.1.j.a.849.1		6
76.47	odd	18		1444.1.j.a.1021.1		6
76.51	even	18		1444.1.j.a.477.1		6
76.55	odd	18		1444.1.j.a.1345.1		6
76.59	even	18		1444.1.j.a.1345.1		6
76.63	odd	18		1444.1.j.a.477.1		6
76.67	even	18		1444.1.j.a.1021.1		6
76.71	even	18		1444.1.j.a.849.1		6
76.75	even	2		76.1.c.a.37.1	✓	1
152.37	odd	2		1216.1.e.b.1025.1		1
152.75	even	2		1216.1.e.a.1025.1		1
228.227	odd	2		684.1.h.a.37.1		1
380.227	odd	4		1900.1.g.a.949.1		2
380.303	odd	4		1900.1.g.a.949.2		2
380.379	even	2		1900.1.e.a.1101.1		1
532.75	odd	6		3724.1.bc.b.949.1		2
532.151	even	6		3724.1.bc.c.569.1		2
532.227	odd	6		3724.1.bc.b.569.1		2
532.303	even	6		3724.1.bc.c.949.1		2
532.531	odd	2		3724.1.e.c.1177.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.1.c.a.37.1	✓	1	4.3	odd	2
76.1.c.a.37.1	✓	1	76.75	even	2
304.1.e.a.113.1		1	1.1	even	1	trivial
304.1.e.a.113.1		1	19.18	odd	2	CM
684.1.h.a.37.1		1	12.11	even	2
684.1.h.a.37.1		1	228.227	odd	2
1216.1.e.a.1025.1		1	8.3	odd	2
1216.1.e.a.1025.1		1	152.75	even	2
1216.1.e.b.1025.1		1	8.5	even	2
1216.1.e.b.1025.1		1	152.37	odd	2
1444.1.h.a.69.1		2	76.11	odd	6
1444.1.h.a.69.1		2	76.27	even	6
1444.1.h.a.293.1		2	76.7	odd	6
1444.1.h.a.293.1		2	76.31	even	6
1444.1.j.a.333.1		6	76.3	even	18
1444.1.j.a.333.1		6	76.35	odd	18
1444.1.j.a.477.1		6	76.51	even	18
1444.1.j.a.477.1		6	76.63	odd	18
1444.1.j.a.849.1		6	76.43	odd	18
1444.1.j.a.849.1		6	76.71	even	18
1444.1.j.a.1021.1		6	76.47	odd	18
1444.1.j.a.1021.1		6	76.67	even	18
1444.1.j.a.1029.1		6	76.15	even	18
1444.1.j.a.1029.1		6	76.23	odd	18
1444.1.j.a.1345.1		6	76.55	odd	18
1444.1.j.a.1345.1		6	76.59	even	18
1900.1.e.a.1101.1		1	20.19	odd	2
1900.1.e.a.1101.1		1	380.379	even	2
1900.1.g.a.949.1		2	20.7	even	4
1900.1.g.a.949.1		2	380.227	odd	4
1900.1.g.a.949.2		2	20.3	even	4
1900.1.g.a.949.2		2	380.303	odd	4
2736.1.o.b.721.1		1	3.2	odd	2
2736.1.o.b.721.1		1	57.56	even	2
3724.1.e.c.1177.1		1	28.27	even	2
3724.1.e.c.1177.1		1	532.531	odd	2
3724.1.bc.b.569.1		2	28.3	even	6
3724.1.bc.b.569.1		2	532.227	odd	6
3724.1.bc.b.949.1		2	28.19	even	6
3724.1.bc.b.949.1		2	532.75	odd	6
3724.1.bc.c.569.1		2	28.11	odd	6
3724.1.bc.c.569.1		2	532.151	even	6
3724.1.bc.c.949.1		2	28.23	odd	6
3724.1.bc.c.949.1		2	532.303	even	6