Embedded newform 1600.2.c.l.449.1

• Label: 1600.2.c.l.449.1
• Level: $1600$
• Weight: $2$
• Character: 1600.449
• Analytic conductor: $12.776$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7760643234\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	1600.449
Dual form			1600.2.c.l.449.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.00000 q^{9} -6.00000i q^{13} -2.00000i q^{17} -10.0000 q^{29} -2.00000i q^{37} +10.0000 q^{41} +7.00000 q^{49} -14.0000i q^{53} +10.0000 q^{61} -6.00000i q^{73} +9.00000 q^{81} -10.0000 q^{89} -18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9} - 20 q^{29} + 20 q^{41} + 14 q^{49} + 20 q^{61} + 18 q^{81} - 20 q^{89}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(1151\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	\(-0.687167\pi\)
			0.554700	−	0.832050i	\(-0.312833\pi\)
\(17\)	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	\(-0.922021\pi\)
			0.970143	−	0.242536i	\(-0.0779791\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	−10.0000	−1.85695	−0.928477	−	0.371391i	\(-0.878881\pi\)
			−0.928477	+	0.371391i	\(0.878881\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	\(-0.947432\pi\)
			0.986394	−	0.164399i	\(-0.0525685\pi\)
\(41\)	10.0000	1.56174	0.780869	−	0.624695i	\(-0.214777\pi\)
			0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)	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	1600.2.c.l.449.1		2
4.3	odd	2	CM	1600.2.c.l.449.1		2
5.2	odd	4		64.2.a.a.1.1		1
5.3	odd	4		1600.2.a.n.1.1		1
5.4	even	2	inner	1600.2.c.l.449.2		2
8.3	odd	2		800.2.c.e.449.2		2
8.5	even	2		800.2.c.e.449.2		2
15.2	even	4		576.2.a.c.1.1		1
20.3	even	4		1600.2.a.n.1.1		1
20.7	even	4		64.2.a.a.1.1		1
20.19	odd	2	inner	1600.2.c.l.449.2		2
24.5	odd	2		7200.2.f.m.6049.2		2
24.11	even	2		7200.2.f.m.6049.2		2
35.27	even	4		3136.2.a.m.1.1		1
40.3	even	4		800.2.a.d.1.1		1
40.13	odd	4		800.2.a.d.1.1		1
40.19	odd	2		800.2.c.e.449.1		2
40.27	even	4		32.2.a.a.1.1	✓	1
40.29	even	2		800.2.c.e.449.1		2
40.37	odd	4		32.2.a.a.1.1	✓	1
55.32	even	4		7744.2.a.v.1.1		1
60.47	odd	4		576.2.a.c.1.1		1
80.27	even	4		256.2.b.b.129.2		2
80.37	odd	4		256.2.b.b.129.2		2
80.67	even	4		256.2.b.b.129.1		2
80.77	odd	4		256.2.b.b.129.1		2
120.29	odd	2		7200.2.f.m.6049.1		2
120.53	even	4		7200.2.a.v.1.1		1
120.59	even	2		7200.2.f.m.6049.1		2
120.77	even	4		288.2.a.d.1.1		1
120.83	odd	4		7200.2.a.v.1.1		1
120.107	odd	4		288.2.a.d.1.1		1
140.27	odd	4		3136.2.a.m.1.1		1
160.27	even	8		1024.2.e.j.769.2		4
160.37	odd	8		1024.2.e.j.769.2		4
160.67	even	8		1024.2.e.j.257.1		4
160.77	odd	8		1024.2.e.j.257.2		4
160.107	even	8		1024.2.e.j.769.1		4
160.117	odd	8		1024.2.e.j.769.1		4
160.147	even	8		1024.2.e.j.257.2		4
160.157	odd	8		1024.2.e.j.257.1		4
220.87	odd	4		7744.2.a.v.1.1		1
240.77	even	4		2304.2.d.j.1153.2		2
240.107	odd	4		2304.2.d.j.1153.1		2
240.197	even	4		2304.2.d.j.1153.1		2
240.227	odd	4		2304.2.d.j.1153.2		2
280.27	odd	4		1568.2.a.e.1.1		1
280.37	odd	12		1568.2.i.g.1537.1		2
280.67	even	12		1568.2.i.g.961.1		2
280.107	even	12		1568.2.i.g.1537.1		2
280.117	even	12		1568.2.i.f.1537.1		2
280.157	even	12		1568.2.i.f.961.1		2
280.187	odd	12		1568.2.i.f.1537.1		2
280.227	odd	12		1568.2.i.f.961.1		2
280.237	even	4		1568.2.a.e.1.1		1
280.277	odd	12		1568.2.i.g.961.1		2
360.67	even	12		2592.2.i.t.865.1		2
360.77	even	12		2592.2.i.e.865.1		2
360.157	odd	12		2592.2.i.t.865.1		2
360.187	even	12		2592.2.i.t.1729.1		2
360.227	odd	12		2592.2.i.e.1729.1		2
360.277	odd	12		2592.2.i.t.1729.1		2
360.317	even	12		2592.2.i.e.1729.1		2
360.347	odd	12		2592.2.i.e.865.1		2
440.197	even	4		3872.2.a.f.1.1		1
440.307	odd	4		3872.2.a.f.1.1		1
520.77	odd	4		5408.2.a.g.1.1		1
520.467	even	4		5408.2.a.g.1.1		1
680.67	even	4		9248.2.a.f.1.1		1
680.237	odd	4		9248.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.a.a.1.1	✓	1	40.27	even	4
32.2.a.a.1.1	✓	1	40.37	odd	4
64.2.a.a.1.1		1	5.2	odd	4
64.2.a.a.1.1		1	20.7	even	4
256.2.b.b.129.1		2	80.67	even	4
256.2.b.b.129.1		2	80.77	odd	4
256.2.b.b.129.2		2	80.27	even	4
256.2.b.b.129.2		2	80.37	odd	4
288.2.a.d.1.1		1	120.77	even	4
288.2.a.d.1.1		1	120.107	odd	4
576.2.a.c.1.1		1	15.2	even	4
576.2.a.c.1.1		1	60.47	odd	4
800.2.a.d.1.1		1	40.3	even	4
800.2.a.d.1.1		1	40.13	odd	4
800.2.c.e.449.1		2	40.19	odd	2
800.2.c.e.449.1		2	40.29	even	2
800.2.c.e.449.2		2	8.3	odd	2
800.2.c.e.449.2		2	8.5	even	2
1024.2.e.j.257.1		4	160.67	even	8
1024.2.e.j.257.1		4	160.157	odd	8
1024.2.e.j.257.2		4	160.77	odd	8
1024.2.e.j.257.2		4	160.147	even	8
1024.2.e.j.769.1		4	160.107	even	8
1024.2.e.j.769.1		4	160.117	odd	8
1024.2.e.j.769.2		4	160.27	even	8
1024.2.e.j.769.2		4	160.37	odd	8
1568.2.a.e.1.1		1	280.27	odd	4
1568.2.a.e.1.1		1	280.237	even	4
1568.2.i.f.961.1		2	280.157	even	12
1568.2.i.f.961.1		2	280.227	odd	12
1568.2.i.f.1537.1		2	280.117	even	12
1568.2.i.f.1537.1		2	280.187	odd	12
1568.2.i.g.961.1		2	280.67	even	12
1568.2.i.g.961.1		2	280.277	odd	12
1568.2.i.g.1537.1		2	280.37	odd	12
1568.2.i.g.1537.1		2	280.107	even	12
1600.2.a.n.1.1		1	5.3	odd	4
1600.2.a.n.1.1		1	20.3	even	4
1600.2.c.l.449.1		2	1.1	even	1	trivial
1600.2.c.l.449.1		2	4.3	odd	2	CM
1600.2.c.l.449.2		2	5.4	even	2	inner
1600.2.c.l.449.2		2	20.19	odd	2	inner
2304.2.d.j.1153.1		2	240.107	odd	4
2304.2.d.j.1153.1		2	240.197	even	4
2304.2.d.j.1153.2		2	240.77	even	4
2304.2.d.j.1153.2		2	240.227	odd	4
2592.2.i.e.865.1		2	360.77	even	12
2592.2.i.e.865.1		2	360.347	odd	12
2592.2.i.e.1729.1		2	360.227	odd	12
2592.2.i.e.1729.1		2	360.317	even	12
2592.2.i.t.865.1		2	360.67	even	12
2592.2.i.t.865.1		2	360.157	odd	12
2592.2.i.t.1729.1		2	360.187	even	12
2592.2.i.t.1729.1		2	360.277	odd	12
3136.2.a.m.1.1		1	35.27	even	4
3136.2.a.m.1.1		1	140.27	odd	4
3872.2.a.f.1.1		1	440.197	even	4
3872.2.a.f.1.1		1	440.307	odd	4
5408.2.a.g.1.1		1	520.77	odd	4
5408.2.a.g.1.1		1	520.467	even	4
7200.2.a.v.1.1		1	120.53	even	4
7200.2.a.v.1.1		1	120.83	odd	4
7200.2.f.m.6049.1		2	120.29	odd	2
7200.2.f.m.6049.1		2	120.59	even	2
7200.2.f.m.6049.2		2	24.5	odd	2
7200.2.f.m.6049.2		2	24.11	even	2
7744.2.a.v.1.1		1	55.32	even	4
7744.2.a.v.1.1		1	220.87	odd	4
9248.2.a.f.1.1		1	680.67	even	4
9248.2.a.f.1.1		1	680.237	odd	4