Embedded newform 576.2.a.c.1.1

• Label: 576.2.a.c.1.1
• Level: $576$
• Weight: $2$
• Character: 576.1
• Self dual: yes
• Analytic conductor: $4.599$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$576 = 2^{6} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	576.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$4.59938315643$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	576.1

$q$-expansion

$f(q)$

$=$

$q-2.00000 q^{5} -6.00000 q^{13} -2.00000 q^{17} -1.00000 q^{25} -10.0000 q^{29} +2.00000 q^{37} -10.0000 q^{41} -7.00000 q^{49} +14.0000 q^{53} +10.0000 q^{61} +12.0000 q^{65} -6.00000 q^{73} +4.00000 q^{85} -10.0000 q^{89} +18.0000 q^{97} +O(q^{100})$

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
$5$	−2.00000	−0.894427				−0.447214	−	0.894427i	$-0.647584\pi$
			−0.447214	+	0.894427i	$0.647584\pi$
$7$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$11$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
			−0.832050	+	0.554700i	$0.812833\pi$
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
			−0.242536	+	0.970143i	$0.577979\pi$
$19$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	−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	$0.5\pi$
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
			0.164399	+	0.986394i	$0.447432\pi$
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
			−0.780869	+	0.624695i	$0.785223\pi$
$43$	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	576.2.a.c.1.1		1
3.2	odd	2		64.2.a.a.1.1		1
4.3	odd	2	CM	576.2.a.c.1.1		1
8.3	odd	2		288.2.a.d.1.1		1
8.5	even	2		288.2.a.d.1.1		1
12.11	even	2		64.2.a.a.1.1		1
15.2	even	4		1600.2.c.l.449.2		2
15.8	even	4		1600.2.c.l.449.1		2
15.14	odd	2		1600.2.a.n.1.1		1
16.3	odd	4		2304.2.d.j.1153.2		2
16.5	even	4		2304.2.d.j.1153.1		2
16.11	odd	4		2304.2.d.j.1153.1		2
16.13	even	4		2304.2.d.j.1153.2		2
21.20	even	2		3136.2.a.m.1.1		1
24.5	odd	2		32.2.a.a.1.1	✓	1
24.11	even	2		32.2.a.a.1.1	✓	1
33.32	even	2		7744.2.a.v.1.1		1
40.3	even	4		7200.2.f.m.6049.2		2
40.13	odd	4		7200.2.f.m.6049.2		2
40.19	odd	2		7200.2.a.v.1.1		1
40.27	even	4		7200.2.f.m.6049.1		2
40.29	even	2		7200.2.a.v.1.1		1
40.37	odd	4		7200.2.f.m.6049.1		2
48.5	odd	4		256.2.b.b.129.2		2
48.11	even	4		256.2.b.b.129.2		2
48.29	odd	4		256.2.b.b.129.1		2
48.35	even	4		256.2.b.b.129.1		2
60.23	odd	4		1600.2.c.l.449.1		2
60.47	odd	4		1600.2.c.l.449.2		2
60.59	even	2		1600.2.a.n.1.1		1
72.5	odd	6		2592.2.i.t.865.1		2
72.11	even	6		2592.2.i.t.1729.1		2
72.13	even	6		2592.2.i.e.865.1		2
72.29	odd	6		2592.2.i.t.1729.1		2
72.43	odd	6		2592.2.i.e.1729.1		2
72.59	even	6		2592.2.i.t.865.1		2
72.61	even	6		2592.2.i.e.1729.1		2
72.67	odd	6		2592.2.i.e.865.1		2
84.83	odd	2		3136.2.a.m.1.1		1
96.5	odd	8		1024.2.e.j.769.2		4
96.11	even	8		1024.2.e.j.769.1		4
96.29	odd	8		1024.2.e.j.257.1		4
96.35	even	8		1024.2.e.j.257.1		4
96.53	odd	8		1024.2.e.j.769.1		4
96.59	even	8		1024.2.e.j.769.2		4
96.77	odd	8		1024.2.e.j.257.2		4
96.83	even	8		1024.2.e.j.257.2		4
120.29	odd	2		800.2.a.d.1.1		1
120.53	even	4		800.2.c.e.449.2		2
120.59	even	2		800.2.a.d.1.1		1
120.77	even	4		800.2.c.e.449.1		2
120.83	odd	4		800.2.c.e.449.2		2
120.107	odd	4		800.2.c.e.449.1		2
132.131	odd	2		7744.2.a.v.1.1		1
168.5	even	6		1568.2.i.f.1537.1		2
168.11	even	6		1568.2.i.g.961.1		2
168.53	odd	6		1568.2.i.g.961.1		2
168.59	odd	6		1568.2.i.f.961.1		2
168.83	odd	2		1568.2.a.e.1.1		1
168.101	even	6		1568.2.i.f.961.1		2
168.107	even	6		1568.2.i.g.1537.1		2
168.125	even	2		1568.2.a.e.1.1		1
168.131	odd	6		1568.2.i.f.1537.1		2
168.149	odd	6		1568.2.i.g.1537.1		2
264.131	odd	2		3872.2.a.f.1.1		1
264.197	even	2		3872.2.a.f.1.1		1
312.77	odd	2		5408.2.a.g.1.1		1
312.155	even	2		5408.2.a.g.1.1		1
408.101	odd	2		9248.2.a.f.1.1		1
408.203	even	2		9248.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.a.a.1.1	✓	1	24.5	odd	2
32.2.a.a.1.1	✓	1	24.11	even	2
64.2.a.a.1.1		1	3.2	odd	2
64.2.a.a.1.1		1	12.11	even	2
256.2.b.b.129.1		2	48.29	odd	4
256.2.b.b.129.1		2	48.35	even	4
256.2.b.b.129.2		2	48.5	odd	4
256.2.b.b.129.2		2	48.11	even	4
288.2.a.d.1.1		1	8.3	odd	2
288.2.a.d.1.1		1	8.5	even	2
576.2.a.c.1.1		1	1.1	even	1	trivial
576.2.a.c.1.1		1	4.3	odd	2	CM
800.2.a.d.1.1		1	120.29	odd	2
800.2.a.d.1.1		1	120.59	even	2
800.2.c.e.449.1		2	120.77	even	4
800.2.c.e.449.1		2	120.107	odd	4
800.2.c.e.449.2		2	120.53	even	4
800.2.c.e.449.2		2	120.83	odd	4
1024.2.e.j.257.1		4	96.29	odd	8
1024.2.e.j.257.1		4	96.35	even	8
1024.2.e.j.257.2		4	96.77	odd	8
1024.2.e.j.257.2		4	96.83	even	8
1024.2.e.j.769.1		4	96.11	even	8
1024.2.e.j.769.1		4	96.53	odd	8
1024.2.e.j.769.2		4	96.5	odd	8
1024.2.e.j.769.2		4	96.59	even	8
1568.2.a.e.1.1		1	168.83	odd	2
1568.2.a.e.1.1		1	168.125	even	2
1568.2.i.f.961.1		2	168.59	odd	6
1568.2.i.f.961.1		2	168.101	even	6
1568.2.i.f.1537.1		2	168.5	even	6
1568.2.i.f.1537.1		2	168.131	odd	6
1568.2.i.g.961.1		2	168.11	even	6
1568.2.i.g.961.1		2	168.53	odd	6
1568.2.i.g.1537.1		2	168.107	even	6
1568.2.i.g.1537.1		2	168.149	odd	6
1600.2.a.n.1.1		1	15.14	odd	2
1600.2.a.n.1.1		1	60.59	even	2
1600.2.c.l.449.1		2	15.8	even	4
1600.2.c.l.449.1		2	60.23	odd	4
1600.2.c.l.449.2		2	15.2	even	4
1600.2.c.l.449.2		2	60.47	odd	4
2304.2.d.j.1153.1		2	16.5	even	4
2304.2.d.j.1153.1		2	16.11	odd	4
2304.2.d.j.1153.2		2	16.3	odd	4
2304.2.d.j.1153.2		2	16.13	even	4
2592.2.i.e.865.1		2	72.13	even	6
2592.2.i.e.865.1		2	72.67	odd	6
2592.2.i.e.1729.1		2	72.43	odd	6
2592.2.i.e.1729.1		2	72.61	even	6
2592.2.i.t.865.1		2	72.5	odd	6
2592.2.i.t.865.1		2	72.59	even	6
2592.2.i.t.1729.1		2	72.11	even	6
2592.2.i.t.1729.1		2	72.29	odd	6
3136.2.a.m.1.1		1	21.20	even	2
3136.2.a.m.1.1		1	84.83	odd	2
3872.2.a.f.1.1		1	264.131	odd	2
3872.2.a.f.1.1		1	264.197	even	2
5408.2.a.g.1.1		1	312.77	odd	2
5408.2.a.g.1.1		1	312.155	even	2
7200.2.a.v.1.1		1	40.19	odd	2
7200.2.a.v.1.1		1	40.29	even	2
7200.2.f.m.6049.1		2	40.27	even	4
7200.2.f.m.6049.1		2	40.37	odd	4
7200.2.f.m.6049.2		2	40.3	even	4
7200.2.f.m.6049.2		2	40.13	odd	4
7744.2.a.v.1.1		1	33.32	even	2
7744.2.a.v.1.1		1	132.131	odd	2
9248.2.a.f.1.1		1	408.101	odd	2
9248.2.a.f.1.1		1	408.203	even	2