Embedded newform 400.3.h.a.399.2

• Label: 400.3.h.a.399.2
• Level: $400$
• Weight: $3$
• Character: 400.399
• Analytic conductor: $10.899$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.8992105744\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2\cdot 5 \)
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			399.2
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	400.399
Dual form			400.3.h.a.399.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-9.00000 q^{9} +10.0000i q^{13} +30.0000i q^{17} -42.0000 q^{29} +70.0000i q^{37} +18.0000 q^{41} -49.0000 q^{49} +90.0000i q^{53} -22.0000 q^{61} -110.000i q^{73} +81.0000 q^{81} +78.0000 q^{89} -130.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 84 q^{29} + 36 q^{41} - 98 q^{49} - 44 q^{61} + 162 q^{81} + 156 q^{89}+O(q^{100}) \)

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		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	0	0
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	10.0000i	0.769231i	0.923077	+	0.384615i	\(0.125666\pi\)
			−0.923077	+	0.384615i	\(0.874334\pi\)
\(17\)	30.0000i	1.76471i	0.470588	+	0.882353i	\(0.344042\pi\)
			−0.470588	+	0.882353i	\(0.655958\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−42.0000	−1.44828	−0.724138	−	0.689655i	\(-0.757762\pi\)
			−0.724138	+	0.689655i	\(0.757762\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	70.0000i	1.89189i	0.324324	+	0.945946i	\(0.394863\pi\)
			−0.324324	+	0.945946i	\(0.605137\pi\)
\(41\)	18.0000	0.439024	0.219512	−	0.975610i	\(-0.429553\pi\)
			0.219512	+	0.975610i	\(0.429553\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.3.h.a.399.2		2
3.2	odd	2		3600.3.j.a.1999.2		2
4.3	odd	2	CM	400.3.h.a.399.2		2
5.2	odd	4		16.3.c.a.15.1	✓	1
5.3	odd	4		400.3.b.a.351.1		1
5.4	even	2	inner	400.3.h.a.399.1		2
8.3	odd	2		1600.3.h.b.1599.1		2
8.5	even	2		1600.3.h.b.1599.1		2
12.11	even	2		3600.3.j.a.1999.2		2
15.2	even	4		144.3.g.a.127.1		1
15.8	even	4		3600.3.e.c.3151.1		1
15.14	odd	2		3600.3.j.a.1999.1		2
20.3	even	4		400.3.b.a.351.1		1
20.7	even	4		16.3.c.a.15.1	✓	1
20.19	odd	2	inner	400.3.h.a.399.1		2
35.2	odd	12		784.3.r.e.655.1		2
35.12	even	12		784.3.r.d.655.1		2
35.17	even	12		784.3.r.d.79.1		2
35.27	even	4		784.3.d.b.687.1		1
35.32	odd	12		784.3.r.e.79.1		2
40.3	even	4		1600.3.b.b.1151.1		1
40.13	odd	4		1600.3.b.b.1151.1		1
40.19	odd	2		1600.3.h.b.1599.2		2
40.27	even	4		64.3.c.a.63.1		1
40.29	even	2		1600.3.h.b.1599.2		2
40.37	odd	4		64.3.c.a.63.1		1
45.2	even	12		1296.3.o.b.271.1		2
45.7	odd	12		1296.3.o.o.271.1		2
45.22	odd	12		1296.3.o.o.703.1		2
45.32	even	12		1296.3.o.b.703.1		2
60.23	odd	4		3600.3.e.c.3151.1		1
60.47	odd	4		144.3.g.a.127.1		1
60.59	even	2		3600.3.j.a.1999.1		2
80.27	even	4		256.3.d.b.127.1		2
80.37	odd	4		256.3.d.b.127.1		2
80.67	even	4		256.3.d.b.127.2		2
80.77	odd	4		256.3.d.b.127.2		2
120.77	even	4		576.3.g.b.127.1		1
120.107	odd	4		576.3.g.b.127.1		1
140.27	odd	4		784.3.d.b.687.1		1
140.47	odd	12		784.3.r.d.655.1		2
140.67	even	12		784.3.r.e.79.1		2
140.87	odd	12		784.3.r.d.79.1		2
140.107	even	12		784.3.r.e.655.1		2
180.7	even	12		1296.3.o.o.271.1		2
180.47	odd	12		1296.3.o.b.271.1		2
180.67	even	12		1296.3.o.o.703.1		2
180.167	odd	12		1296.3.o.b.703.1		2
240.77	even	4		2304.3.b.f.127.1		2
240.107	odd	4		2304.3.b.f.127.2		2
240.197	even	4		2304.3.b.f.127.2		2
240.227	odd	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	5.2	odd	4
16.3.c.a.15.1	✓	1	20.7	even	4
64.3.c.a.63.1		1	40.27	even	4
64.3.c.a.63.1		1	40.37	odd	4
144.3.g.a.127.1		1	15.2	even	4
144.3.g.a.127.1		1	60.47	odd	4
256.3.d.b.127.1		2	80.27	even	4
256.3.d.b.127.1		2	80.37	odd	4
256.3.d.b.127.2		2	80.67	even	4
256.3.d.b.127.2		2	80.77	odd	4
400.3.b.a.351.1		1	5.3	odd	4
400.3.b.a.351.1		1	20.3	even	4
400.3.h.a.399.1		2	5.4	even	2	inner
400.3.h.a.399.1		2	20.19	odd	2	inner
400.3.h.a.399.2		2	1.1	even	1	trivial
400.3.h.a.399.2		2	4.3	odd	2	CM
576.3.g.b.127.1		1	120.77	even	4
576.3.g.b.127.1		1	120.107	odd	4
784.3.d.b.687.1		1	35.27	even	4
784.3.d.b.687.1		1	140.27	odd	4
784.3.r.d.79.1		2	35.17	even	12
784.3.r.d.79.1		2	140.87	odd	12
784.3.r.d.655.1		2	35.12	even	12
784.3.r.d.655.1		2	140.47	odd	12
784.3.r.e.79.1		2	35.32	odd	12
784.3.r.e.79.1		2	140.67	even	12
784.3.r.e.655.1		2	35.2	odd	12
784.3.r.e.655.1		2	140.107	even	12
1296.3.o.b.271.1		2	45.2	even	12
1296.3.o.b.271.1		2	180.47	odd	12
1296.3.o.b.703.1		2	45.32	even	12
1296.3.o.b.703.1		2	180.167	odd	12
1296.3.o.o.271.1		2	45.7	odd	12
1296.3.o.o.271.1		2	180.7	even	12
1296.3.o.o.703.1		2	45.22	odd	12
1296.3.o.o.703.1		2	180.67	even	12
1600.3.b.b.1151.1		1	40.3	even	4
1600.3.b.b.1151.1		1	40.13	odd	4
1600.3.h.b.1599.1		2	8.3	odd	2
1600.3.h.b.1599.1		2	8.5	even	2
1600.3.h.b.1599.2		2	40.19	odd	2
1600.3.h.b.1599.2		2	40.29	even	2
2304.3.b.f.127.1		2	240.77	even	4
2304.3.b.f.127.1		2	240.227	odd	4
2304.3.b.f.127.2		2	240.107	odd	4
2304.3.b.f.127.2		2	240.197	even	4
3600.3.e.c.3151.1		1	15.8	even	4
3600.3.e.c.3151.1		1	60.23	odd	4
3600.3.j.a.1999.1		2	15.14	odd	2
3600.3.j.a.1999.1		2	60.59	even	2
3600.3.j.a.1999.2		2	3.2	odd	2
3600.3.j.a.1999.2		2	12.11	even	2