Embedded newform 1600.2.f.j.1249.3

• Label: 1600.2.f.j.1249.3
• Level: $1600$
• Weight: $2$
• Character: 1600.1249
• 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.f (of order \(2\), degree \(1\), not 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:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1249.3
Root			\(-1.22474 + 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	1600.1249
Dual form			1600.2.f.j.1249.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.44949 q^{3} +8.89898 q^{9} -5.44949i q^{11} -1.89898i q^{17} +6.34847i q^{19} +20.3485 q^{27} -18.7980i q^{33} -6.79796 q^{41} -10.0000 q^{43} +7.00000 q^{49} -6.55051i q^{51} +21.8990i q^{57} -6.00000i q^{59} -0.348469 q^{67} +15.6969i q^{73} +43.4949 q^{81} +6.55051 q^{83} +4.10102 q^{89} +10.0000i q^{97} -48.4949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 16 q^{9} + 52 q^{27} + 12 q^{41} - 40 q^{43} + 28 q^{49} + 28 q^{67} + 76 q^{81} + 36 q^{83} + 36 q^{89}+O(q^{100}) \)

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	3.44949	1.99156	0.995782	−	0.0917517i	\(-0.0292466\pi\)
0.995782	+	0.0917517i				\(0.0292466\pi\)
\(5\)	0	0
			\(7\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(11\)	− 5.44949i	− 1.64308i	−0.570149	−	0.821541i	\(-0.693114\pi\)
			0.570149	−	0.821541i	\(-0.306886\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	− 1.89898i	− 0.460570i	−0.973123	−	0.230285i	\(-0.926034\pi\)
			0.973123	−	0.230285i	\(-0.0739659\pi\)
\(19\)	6.34847i	1.45644i	0.685344	+	0.728219i	\(0.259652\pi\)
			−0.685344	+	0.728219i	\(0.740348\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	−6.79796	−1.06166	−0.530831	−	0.847477i	\(-0.678120\pi\)
			−0.530831	+	0.847477i	\(0.678120\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(59\)	− 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			\(\pi\)
\(67\)	−0.348469	−0.0425723	−0.0212861	−	0.999773i	\(-0.506776\pi\)
			−0.0212861	+	0.999773i	\(0.506776\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	15.6969i	1.83719i	0.395203	+	0.918594i	\(0.370674\pi\)
			−0.395203	+	0.918594i	\(0.629326\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	6.55051	0.719012	0.359506	−	0.933143i	\(-0.382945\pi\)
			0.359506	+	0.933143i	\(0.382945\pi\)
\(89\)	4.10102	0.434707	0.217354	−	0.976093i	\(-0.430258\pi\)
			0.217354	+	0.976093i	\(0.430258\pi\)
\(97\)	10.0000i	1.01535i	0.861550	+	0.507673i	\(0.169494\pi\)
			−0.861550	+	0.507673i	\(0.830506\pi\)

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.f.j.1249.3		4
4.3	odd	2		1600.2.f.f.1249.2		4
5.2	odd	4		1600.2.d.c.801.1	✓	4
5.3	odd	4		1600.2.d.d.801.4	yes	4
5.4	even	2		1600.2.f.f.1249.1		4
8.3	odd	2	CM	1600.2.f.j.1249.3		4
8.5	even	2		1600.2.f.f.1249.2		4
20.3	even	4		1600.2.d.d.801.1	yes	4
20.7	even	4		1600.2.d.c.801.4	yes	4
20.19	odd	2	inner	1600.2.f.j.1249.4		4
40.3	even	4		1600.2.d.d.801.4	yes	4
40.13	odd	4		1600.2.d.d.801.1	yes	4
40.19	odd	2		1600.2.f.f.1249.1		4
40.27	even	4		1600.2.d.c.801.1	✓	4
40.29	even	2	inner	1600.2.f.j.1249.4		4
40.37	odd	4		1600.2.d.c.801.4	yes	4
80.3	even	4		6400.2.a.ci.1.2		2
80.13	odd	4		6400.2.a.bc.1.1		2
80.27	even	4		6400.2.a.ch.1.2		2
80.37	odd	4		6400.2.a.bd.1.1		2
80.43	even	4		6400.2.a.bc.1.1		2
80.53	odd	4		6400.2.a.ci.1.2		2
80.67	even	4		6400.2.a.bd.1.1		2
80.77	odd	4		6400.2.a.ch.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.c.801.1	✓	4	5.2	odd	4
1600.2.d.c.801.1	✓	4	40.27	even	4
1600.2.d.c.801.4	yes	4	20.7	even	4
1600.2.d.c.801.4	yes	4	40.37	odd	4
1600.2.d.d.801.1	yes	4	20.3	even	4
1600.2.d.d.801.1	yes	4	40.13	odd	4
1600.2.d.d.801.4	yes	4	5.3	odd	4
1600.2.d.d.801.4	yes	4	40.3	even	4
1600.2.f.f.1249.1		4	5.4	even	2
1600.2.f.f.1249.1		4	40.19	odd	2
1600.2.f.f.1249.2		4	4.3	odd	2
1600.2.f.f.1249.2		4	8.5	even	2
1600.2.f.j.1249.3		4	1.1	even	1	trivial
1600.2.f.j.1249.3		4	8.3	odd	2	CM
1600.2.f.j.1249.4		4	20.19	odd	2	inner
1600.2.f.j.1249.4		4	40.29	even	2	inner
6400.2.a.bc.1.1		2	80.13	odd	4
6400.2.a.bc.1.1		2	80.43	even	4
6400.2.a.bd.1.1		2	80.37	odd	4
6400.2.a.bd.1.1		2	80.67	even	4
6400.2.a.ch.1.2		2	80.27	even	4
6400.2.a.ch.1.2		2	80.77	odd	4
6400.2.a.ci.1.2		2	80.3	even	4
6400.2.a.ci.1.2		2	80.53	odd	4