Embedded newform 400.3.b.a.351.1

• Label: 400.3.b.a.351.1
• Level: $400$
• Weight: $3$
• Character: 400.351
• Self dual: yes
• Analytic conductor: $10.899$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.8992105744\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			351.1
Character	\(\chi\)	\(=\)	400.351

$q$-expansion

\(f(q)\)

\(=\)

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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						\(\pi\)
\(5\)	0	0
			\(7\)	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	\(-0.625666\pi\)
			−0.384615	+	0.923077i	\(0.625666\pi\)
\(17\)	30.0000	1.76471	0.882353	−	0.470588i	\(-0.155958\pi\)
			0.882353	+	0.470588i	\(0.155958\pi\)
\(19\)	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	\(-0.105137\pi\)
			0.945946	+	0.324324i	\(0.105137\pi\)
\(41\)	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	400.3.b.a.351.1		1
3.2	odd	2		3600.3.e.c.3151.1		1
4.3	odd	2	CM	400.3.b.a.351.1		1
5.2	odd	4		400.3.h.a.399.2		2
5.3	odd	4		400.3.h.a.399.1		2
5.4	even	2		16.3.c.a.15.1	✓	1
8.3	odd	2		1600.3.b.b.1151.1		1
8.5	even	2		1600.3.b.b.1151.1		1
12.11	even	2		3600.3.e.c.3151.1		1
15.2	even	4		3600.3.j.a.1999.2		2
15.8	even	4		3600.3.j.a.1999.1		2
15.14	odd	2		144.3.g.a.127.1		1
20.3	even	4		400.3.h.a.399.1		2
20.7	even	4		400.3.h.a.399.2		2
20.19	odd	2		16.3.c.a.15.1	✓	1
35.4	even	6		784.3.r.e.79.1		2
35.9	even	6		784.3.r.e.655.1		2
35.19	odd	6		784.3.r.d.655.1		2
35.24	odd	6		784.3.r.d.79.1		2
35.34	odd	2		784.3.d.b.687.1		1
40.3	even	4		1600.3.h.b.1599.2		2
40.13	odd	4		1600.3.h.b.1599.2		2
40.19	odd	2		64.3.c.a.63.1		1
40.27	even	4		1600.3.h.b.1599.1		2
40.29	even	2		64.3.c.a.63.1		1
40.37	odd	4		1600.3.h.b.1599.1		2
45.4	even	6		1296.3.o.o.703.1		2
45.14	odd	6		1296.3.o.b.703.1		2
45.29	odd	6		1296.3.o.b.271.1		2
45.34	even	6		1296.3.o.o.271.1		2
60.23	odd	4		3600.3.j.a.1999.1		2
60.47	odd	4		3600.3.j.a.1999.2		2
60.59	even	2		144.3.g.a.127.1		1
80.19	odd	4		256.3.d.b.127.2		2
80.29	even	4		256.3.d.b.127.2		2
80.59	odd	4		256.3.d.b.127.1		2
80.69	even	4		256.3.d.b.127.1		2
120.29	odd	2		576.3.g.b.127.1		1
120.59	even	2		576.3.g.b.127.1		1
140.19	even	6		784.3.r.d.655.1		2
140.39	odd	6		784.3.r.e.79.1		2
140.59	even	6		784.3.r.d.79.1		2
140.79	odd	6		784.3.r.e.655.1		2
140.139	even	2		784.3.d.b.687.1		1
180.59	even	6		1296.3.o.b.703.1		2
180.79	odd	6		1296.3.o.o.271.1		2
180.119	even	6		1296.3.o.b.271.1		2
180.139	odd	6		1296.3.o.o.703.1		2
240.29	odd	4		2304.3.b.f.127.1		2
240.59	even	4		2304.3.b.f.127.2		2
240.149	odd	4		2304.3.b.f.127.2		2
240.179	even	4		2304.3.b.f.127.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.c.a.15.1	✓	1	5.4	even	2
16.3.c.a.15.1	✓	1	20.19	odd	2
64.3.c.a.63.1		1	40.19	odd	2
64.3.c.a.63.1		1	40.29	even	2
144.3.g.a.127.1		1	15.14	odd	2
144.3.g.a.127.1		1	60.59	even	2
256.3.d.b.127.1		2	80.59	odd	4
256.3.d.b.127.1		2	80.69	even	4
256.3.d.b.127.2		2	80.19	odd	4
256.3.d.b.127.2		2	80.29	even	4
400.3.b.a.351.1		1	1.1	even	1	trivial
400.3.b.a.351.1		1	4.3	odd	2	CM
400.3.h.a.399.1		2	5.3	odd	4
400.3.h.a.399.1		2	20.3	even	4
400.3.h.a.399.2		2	5.2	odd	4
400.3.h.a.399.2		2	20.7	even	4
576.3.g.b.127.1		1	120.29	odd	2
576.3.g.b.127.1		1	120.59	even	2
784.3.d.b.687.1		1	35.34	odd	2
784.3.d.b.687.1		1	140.139	even	2
784.3.r.d.79.1		2	35.24	odd	6
784.3.r.d.79.1		2	140.59	even	6
784.3.r.d.655.1		2	35.19	odd	6
784.3.r.d.655.1		2	140.19	even	6
784.3.r.e.79.1		2	35.4	even	6
784.3.r.e.79.1		2	140.39	odd	6
784.3.r.e.655.1		2	35.9	even	6
784.3.r.e.655.1		2	140.79	odd	6
1296.3.o.b.271.1		2	45.29	odd	6
1296.3.o.b.271.1		2	180.119	even	6
1296.3.o.b.703.1		2	45.14	odd	6
1296.3.o.b.703.1		2	180.59	even	6
1296.3.o.o.271.1		2	45.34	even	6
1296.3.o.o.271.1		2	180.79	odd	6
1296.3.o.o.703.1		2	45.4	even	6
1296.3.o.o.703.1		2	180.139	odd	6
1600.3.b.b.1151.1		1	8.3	odd	2
1600.3.b.b.1151.1		1	8.5	even	2
1600.3.h.b.1599.1		2	40.27	even	4
1600.3.h.b.1599.1		2	40.37	odd	4
1600.3.h.b.1599.2		2	40.3	even	4
1600.3.h.b.1599.2		2	40.13	odd	4
2304.3.b.f.127.1		2	240.29	odd	4
2304.3.b.f.127.1		2	240.179	even	4
2304.3.b.f.127.2		2	240.59	even	4
2304.3.b.f.127.2		2	240.149	odd	4
3600.3.e.c.3151.1		1	3.2	odd	2
3600.3.e.c.3151.1		1	12.11	even	2
3600.3.j.a.1999.1		2	15.8	even	4
3600.3.j.a.1999.1		2	60.23	odd	4
3600.3.j.a.1999.2		2	15.2	even	4
3600.3.j.a.1999.2		2	60.47	odd	4