Embedded newform 16.3.c.a.15.1

• Label: 16.3.c.a.15.1
• Level: $16$
• Weight: $3$
• Character: 16.15
• Self dual: yes
• Analytic conductor: $0.436$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.435968422976\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			15.1
Character	\(\chi\)	\(=\)	16.15

$q$-expansion

\(f(q)\)

\(=\)

\(q-6.00000 q^{5} +9.00000 q^{9} +O(q^{10})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(1\)	\(-1\)

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						\(\pi\)
\(5\)	−6.00000	−1.20000	−0.600000	−	0.800000i	\(-0.704833\pi\)
			−0.600000	+	0.800000i	\(0.704833\pi\)
\(7\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	10.0000	0.769231	0.384615	−	0.923077i	\(-0.374334\pi\)
			0.384615	+	0.923077i	\(0.374334\pi\)
\(17\)	−30.0000	−1.76471	−0.882353	−	0.470588i	\(-0.844042\pi\)
			−0.882353	+	0.470588i	\(0.844042\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	42.0000	1.44828	0.724138	−	0.689655i	\(-0.242238\pi\)
			0.724138	+	0.689655i	\(0.242238\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	−70.0000	−1.89189	−0.945946	−	0.324324i	\(-0.894863\pi\)
			−0.945946	+	0.324324i	\(0.894863\pi\)
\(41\)	18.0000	0.439024	0.219512	−	0.975610i	\(-0.429553\pi\)
			0.219512	+	0.975610i	\(0.429553\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.3.c.a.15.1	✓	1
3.2	odd	2		144.3.g.a.127.1		1
4.3	odd	2	CM	16.3.c.a.15.1	✓	1
5.2	odd	4		400.3.h.a.399.1		2
5.3	odd	4		400.3.h.a.399.2		2
5.4	even	2		400.3.b.a.351.1		1
7.2	even	3		784.3.r.e.655.1		2
7.3	odd	6		784.3.r.d.79.1		2
7.4	even	3		784.3.r.e.79.1		2
7.5	odd	6		784.3.r.d.655.1		2
7.6	odd	2		784.3.d.b.687.1		1
8.3	odd	2		64.3.c.a.63.1		1
8.5	even	2		64.3.c.a.63.1		1
9.2	odd	6		1296.3.o.b.271.1		2
9.4	even	3		1296.3.o.o.703.1		2
9.5	odd	6		1296.3.o.b.703.1		2
9.7	even	3		1296.3.o.o.271.1		2
12.11	even	2		144.3.g.a.127.1		1
15.2	even	4		3600.3.j.a.1999.1		2
15.8	even	4		3600.3.j.a.1999.2		2
15.14	odd	2		3600.3.e.c.3151.1		1
16.3	odd	4		256.3.d.b.127.2		2
16.5	even	4		256.3.d.b.127.1		2
16.11	odd	4		256.3.d.b.127.1		2
16.13	even	4		256.3.d.b.127.2		2
20.3	even	4		400.3.h.a.399.2		2
20.7	even	4		400.3.h.a.399.1		2
20.19	odd	2		400.3.b.a.351.1		1
24.5	odd	2		576.3.g.b.127.1		1
24.11	even	2		576.3.g.b.127.1		1
28.3	even	6		784.3.r.d.79.1		2
28.11	odd	6		784.3.r.e.79.1		2
28.19	even	6		784.3.r.d.655.1		2
28.23	odd	6		784.3.r.e.655.1		2
28.27	even	2		784.3.d.b.687.1		1
36.7	odd	6		1296.3.o.o.271.1		2
36.11	even	6		1296.3.o.b.271.1		2
36.23	even	6		1296.3.o.b.703.1		2
36.31	odd	6		1296.3.o.o.703.1		2
40.3	even	4		1600.3.h.b.1599.1		2
40.13	odd	4		1600.3.h.b.1599.1		2
40.19	odd	2		1600.3.b.b.1151.1		1
40.27	even	4		1600.3.h.b.1599.2		2
40.29	even	2		1600.3.b.b.1151.1		1
40.37	odd	4		1600.3.h.b.1599.2		2
48.5	odd	4		2304.3.b.f.127.2		2
48.11	even	4		2304.3.b.f.127.2		2
48.29	odd	4		2304.3.b.f.127.1		2
48.35	even	4		2304.3.b.f.127.1		2
60.23	odd	4		3600.3.j.a.1999.2		2
60.47	odd	4		3600.3.j.a.1999.1		2
60.59	even	2		3600.3.e.c.3151.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.c.a.15.1	✓	1	1.1	even	1	trivial
16.3.c.a.15.1	✓	1	4.3	odd	2	CM
64.3.c.a.63.1		1	8.3	odd	2
64.3.c.a.63.1		1	8.5	even	2
144.3.g.a.127.1		1	3.2	odd	2
144.3.g.a.127.1		1	12.11	even	2
256.3.d.b.127.1		2	16.5	even	4
256.3.d.b.127.1		2	16.11	odd	4
256.3.d.b.127.2		2	16.3	odd	4
256.3.d.b.127.2		2	16.13	even	4
400.3.b.a.351.1		1	5.4	even	2
400.3.b.a.351.1		1	20.19	odd	2
400.3.h.a.399.1		2	5.2	odd	4
400.3.h.a.399.1		2	20.7	even	4
400.3.h.a.399.2		2	5.3	odd	4
400.3.h.a.399.2		2	20.3	even	4
576.3.g.b.127.1		1	24.5	odd	2
576.3.g.b.127.1		1	24.11	even	2
784.3.d.b.687.1		1	7.6	odd	2
784.3.d.b.687.1		1	28.27	even	2
784.3.r.d.79.1		2	7.3	odd	6
784.3.r.d.79.1		2	28.3	even	6
784.3.r.d.655.1		2	7.5	odd	6
784.3.r.d.655.1		2	28.19	even	6
784.3.r.e.79.1		2	7.4	even	3
784.3.r.e.79.1		2	28.11	odd	6
784.3.r.e.655.1		2	7.2	even	3
784.3.r.e.655.1		2	28.23	odd	6
1296.3.o.b.271.1		2	9.2	odd	6
1296.3.o.b.271.1		2	36.11	even	6
1296.3.o.b.703.1		2	9.5	odd	6
1296.3.o.b.703.1		2	36.23	even	6
1296.3.o.o.271.1		2	9.7	even	3
1296.3.o.o.271.1		2	36.7	odd	6
1296.3.o.o.703.1		2	9.4	even	3
1296.3.o.o.703.1		2	36.31	odd	6
1600.3.b.b.1151.1		1	40.19	odd	2
1600.3.b.b.1151.1		1	40.29	even	2
1600.3.h.b.1599.1		2	40.3	even	4
1600.3.h.b.1599.1		2	40.13	odd	4
1600.3.h.b.1599.2		2	40.27	even	4
1600.3.h.b.1599.2		2	40.37	odd	4
2304.3.b.f.127.1		2	48.29	odd	4
2304.3.b.f.127.1		2	48.35	even	4
2304.3.b.f.127.2		2	48.5	odd	4
2304.3.b.f.127.2		2	48.11	even	4
3600.3.e.c.3151.1		1	15.14	odd	2
3600.3.e.c.3151.1		1	60.59	even	2
3600.3.j.a.1999.1		2	15.2	even	4
3600.3.j.a.1999.1		2	60.47	odd	4
3600.3.j.a.1999.2		2	15.8	even	4
3600.3.j.a.1999.2		2	60.23	odd	4