Embedded newform 576.2.d.a.289.1

• Label: 576.2.d.a.289.1
• Level: $576$
• Weight: $2$
• Character: 576.289
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			289.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	576.289
Dual form			576.2.d.a.289.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-6.00000i q^{11} +6.00000 q^{17} -2.00000i q^{19} +5.00000 q^{25} +6.00000 q^{41} -10.0000i q^{43} -7.00000 q^{49} -6.00000i q^{59} +14.0000i q^{67} +2.00000 q^{73} +18.0000i q^{83} -18.0000 q^{89} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{17} + 10 q^{25} + 12 q^{41} - 14 q^{49} + 4 q^{73} - 36 q^{89} + 20 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	\(-0.640219\pi\)
			0.426401	−	0.904534i	\(-0.359781\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	\(-0.926318\pi\)
			0.973329	−	0.229416i	\(-0.0736815\pi\)
\(23\)	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			\(\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	− 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	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.d.a.289.1		2
3.2	odd	2		64.2.b.a.33.1	✓	2
4.3	odd	2	inner	576.2.d.a.289.2		2
8.3	odd	2	CM	576.2.d.a.289.1		2
8.5	even	2	inner	576.2.d.a.289.2		2
12.11	even	2		64.2.b.a.33.2	yes	2
15.2	even	4		1600.2.f.a.1249.2		2
15.8	even	4		1600.2.f.b.1249.2		2
15.14	odd	2		1600.2.d.a.801.2		2
16.3	odd	4		2304.2.a.i.1.1		1
16.5	even	4		2304.2.a.i.1.1		1
16.11	odd	4		2304.2.a.h.1.1		1
16.13	even	4		2304.2.a.h.1.1		1
21.20	even	2		3136.2.b.b.1569.2		2
24.5	odd	2		64.2.b.a.33.2	yes	2
24.11	even	2		64.2.b.a.33.1	✓	2
48.5	odd	4		256.2.a.a.1.1		1
48.11	even	4		256.2.a.d.1.1		1
48.29	odd	4		256.2.a.d.1.1		1
48.35	even	4		256.2.a.a.1.1		1
60.23	odd	4		1600.2.f.a.1249.1		2
60.47	odd	4		1600.2.f.b.1249.1		2
60.59	even	2		1600.2.d.a.801.1		2
84.83	odd	2		3136.2.b.b.1569.1		2
96.5	odd	8		1024.2.e.l.257.2		4
96.11	even	8		1024.2.e.l.257.2		4
96.29	odd	8		1024.2.e.l.769.2		4
96.35	even	8		1024.2.e.l.769.1		4
96.53	odd	8		1024.2.e.l.257.1		4
96.59	even	8		1024.2.e.l.257.1		4
96.77	odd	8		1024.2.e.l.769.1		4
96.83	even	8		1024.2.e.l.769.2		4
120.29	odd	2		1600.2.d.a.801.1		2
120.53	even	4		1600.2.f.a.1249.1		2
120.59	even	2		1600.2.d.a.801.2		2
120.77	even	4		1600.2.f.b.1249.1		2
120.83	odd	4		1600.2.f.b.1249.2		2
120.107	odd	4		1600.2.f.a.1249.2		2
168.83	odd	2		3136.2.b.b.1569.2		2
168.125	even	2		3136.2.b.b.1569.1		2
240.29	odd	4		6400.2.a.a.1.1		1
240.59	even	4		6400.2.a.a.1.1		1
240.149	odd	4		6400.2.a.x.1.1		1
240.179	even	4		6400.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.2.b.a.33.1	✓	2	3.2	odd	2
64.2.b.a.33.1	✓	2	24.11	even	2
64.2.b.a.33.2	yes	2	12.11	even	2
64.2.b.a.33.2	yes	2	24.5	odd	2
256.2.a.a.1.1		1	48.5	odd	4
256.2.a.a.1.1		1	48.35	even	4
256.2.a.d.1.1		1	48.11	even	4
256.2.a.d.1.1		1	48.29	odd	4
576.2.d.a.289.1		2	1.1	even	1	trivial
576.2.d.a.289.1		2	8.3	odd	2	CM
576.2.d.a.289.2		2	4.3	odd	2	inner
576.2.d.a.289.2		2	8.5	even	2	inner
1024.2.e.l.257.1		4	96.53	odd	8
1024.2.e.l.257.1		4	96.59	even	8
1024.2.e.l.257.2		4	96.5	odd	8
1024.2.e.l.257.2		4	96.11	even	8
1024.2.e.l.769.1		4	96.35	even	8
1024.2.e.l.769.1		4	96.77	odd	8
1024.2.e.l.769.2		4	96.29	odd	8
1024.2.e.l.769.2		4	96.83	even	8
1600.2.d.a.801.1		2	60.59	even	2
1600.2.d.a.801.1		2	120.29	odd	2
1600.2.d.a.801.2		2	15.14	odd	2
1600.2.d.a.801.2		2	120.59	even	2
1600.2.f.a.1249.1		2	60.23	odd	4
1600.2.f.a.1249.1		2	120.53	even	4
1600.2.f.a.1249.2		2	15.2	even	4
1600.2.f.a.1249.2		2	120.107	odd	4
1600.2.f.b.1249.1		2	60.47	odd	4
1600.2.f.b.1249.1		2	120.77	even	4
1600.2.f.b.1249.2		2	15.8	even	4
1600.2.f.b.1249.2		2	120.83	odd	4
2304.2.a.h.1.1		1	16.11	odd	4
2304.2.a.h.1.1		1	16.13	even	4
2304.2.a.i.1.1		1	16.3	odd	4
2304.2.a.i.1.1		1	16.5	even	4
3136.2.b.b.1569.1		2	84.83	odd	2
3136.2.b.b.1569.1		2	168.125	even	2
3136.2.b.b.1569.2		2	21.20	even	2
3136.2.b.b.1569.2		2	168.83	odd	2
6400.2.a.a.1.1		1	240.29	odd	4
6400.2.a.a.1.1		1	240.59	even	4
6400.2.a.x.1.1		1	240.149	odd	4
6400.2.a.x.1.1		1	240.179	even	4