Embedded newform 1600.2.d.c.801.2

• Label: 1600.2.d.c.801.2
• Level: $1600$
• Weight: $2$
• Character: 1600.801
• Analytic conductor: $12.776$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7760643234\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{6})\)
Defining polynomial:	\( x^{4} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			801.2
Root			\(-1.22474 - 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	1600.801
Dual form			1600.2.d.c.801.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.44949i q^{3} +0.898979 q^{9} +0.550510i q^{11} -7.89898 q^{17} -8.34847i q^{19} -5.65153i q^{27} +0.797959 q^{33} +12.7980 q^{41} -10.0000i q^{43} -7.00000 q^{49} +11.4495i q^{51} -12.1010 q^{57} -6.00000i q^{59} -14.3485i q^{67} -13.6969 q^{73} -5.49490 q^{81} +11.4495i q^{83} -13.8990 q^{89} -10.0000 q^{97} +0.494897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{9} - 12 q^{17} - 36 q^{33} + 12 q^{41} - 28 q^{49} - 68 q^{57} + 4 q^{73} + 76 q^{81} - 36 q^{89} - 40 q^{97}+O(q^{100}) \)

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	− 1.44949i	− 0.836863i	−0.908248	−	0.418432i	\(-0.862580\pi\)
0.908248	−	0.418432i				\(-0.137420\pi\)
\(5\)	0	0
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	0.550510i	0.165985i	0.996550	+	0.0829925i	\(0.0264478\pi\)
			−0.996550	+	0.0829925i	\(0.973552\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−7.89898	−1.91578	−0.957892	−	0.287129i	\(-0.907299\pi\)
			−0.957892	+	0.287129i	\(0.907299\pi\)
\(19\)	− 8.34847i	− 1.91527i	−0.287984	−	0.957635i	\(-0.592985\pi\)
			0.287984	−	0.957635i	\(-0.407015\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	12.7980	1.99871	0.999353	−	0.0359748i	\(-0.0114536\pi\)
			0.999353	+	0.0359748i	\(0.0114536\pi\)
\(43\)	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	\(-0.723975\pi\)
			0.646997	−	0.762493i	\(-0.276025\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	\(-0.872279\pi\)
			0.920575	−	0.390567i	\(-0.127721\pi\)
\(61\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(67\)	− 14.3485i	− 1.75294i	−0.481452	−	0.876472i	\(-0.659891\pi\)
			0.481452	−	0.876472i	\(-0.340109\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−13.6969	−1.60311	−0.801553	−	0.597924i	\(-0.795992\pi\)
			−0.801553	+	0.597924i	\(0.795992\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	11.4495i	1.25674i	0.777913	+	0.628372i	\(0.216279\pi\)
			−0.777913	+	0.628372i	\(0.783721\pi\)
\(89\)	−13.8990	−1.47329	−0.736644	−	0.676280i	\(-0.763591\pi\)
			−0.736644	+	0.676280i	\(0.763591\pi\)
\(97\)	−10.0000	−1.01535	−0.507673	−	0.861550i	\(-0.669494\pi\)
			−0.507673	+	0.861550i	\(0.669494\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.c.801.2	✓	4
4.3	odd	2	inner	1600.2.d.c.801.3	yes	4
5.2	odd	4		1600.2.f.j.1249.2		4
5.3	odd	4		1600.2.f.f.1249.4		4
5.4	even	2		1600.2.d.d.801.3	yes	4
8.3	odd	2	CM	1600.2.d.c.801.2	✓	4
8.5	even	2	inner	1600.2.d.c.801.3	yes	4
16.3	odd	4		6400.2.a.ch.1.1		2
16.5	even	4		6400.2.a.ch.1.1		2
16.11	odd	4		6400.2.a.bd.1.2		2
16.13	even	4		6400.2.a.bd.1.2		2
20.3	even	4		1600.2.f.j.1249.1		4
20.7	even	4		1600.2.f.f.1249.3		4
20.19	odd	2		1600.2.d.d.801.2	yes	4
40.3	even	4		1600.2.f.f.1249.4		4
40.13	odd	4		1600.2.f.j.1249.1		4
40.19	odd	2		1600.2.d.d.801.3	yes	4
40.27	even	4		1600.2.f.j.1249.2		4
40.29	even	2		1600.2.d.d.801.2	yes	4
40.37	odd	4		1600.2.f.f.1249.3		4
80.19	odd	4		6400.2.a.bc.1.2		2
80.29	even	4		6400.2.a.ci.1.1		2
80.59	odd	4		6400.2.a.ci.1.1		2
80.69	even	4		6400.2.a.bc.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.c.801.2	✓	4	1.1	even	1	trivial
1600.2.d.c.801.2	✓	4	8.3	odd	2	CM
1600.2.d.c.801.3	yes	4	4.3	odd	2	inner
1600.2.d.c.801.3	yes	4	8.5	even	2	inner
1600.2.d.d.801.2	yes	4	20.19	odd	2
1600.2.d.d.801.2	yes	4	40.29	even	2
1600.2.d.d.801.3	yes	4	5.4	even	2
1600.2.d.d.801.3	yes	4	40.19	odd	2
1600.2.f.f.1249.3		4	20.7	even	4
1600.2.f.f.1249.3		4	40.37	odd	4
1600.2.f.f.1249.4		4	5.3	odd	4
1600.2.f.f.1249.4		4	40.3	even	4
1600.2.f.j.1249.1		4	20.3	even	4
1600.2.f.j.1249.1		4	40.13	odd	4
1600.2.f.j.1249.2		4	5.2	odd	4
1600.2.f.j.1249.2		4	40.27	even	4
6400.2.a.bc.1.2		2	80.19	odd	4
6400.2.a.bc.1.2		2	80.69	even	4
6400.2.a.bd.1.2		2	16.11	odd	4
6400.2.a.bd.1.2		2	16.13	even	4
6400.2.a.ch.1.1		2	16.3	odd	4
6400.2.a.ch.1.1		2	16.5	even	4
6400.2.a.ci.1.1		2	80.29	even	4
6400.2.a.ci.1.1		2	80.59	odd	4