Embedded newform 576.8.a.f.1.1

• Label: 576.8.a.f.1.1
• Level: $576$
• Weight: $8$
• Character: 576.1
• Self dual: yes
• Analytic conductor: $179.934$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$576 = 2^{6} \cdot 3^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	576.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$179.933774679$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

$f(q)$

$=$

$q-210.000 q^{5} -1016.00 q^{7} -1092.00 q^{11} -1382.00 q^{13} -14706.0 q^{17} -39940.0 q^{19} +68712.0 q^{23} -34025.0 q^{25} -102570. q^{29} -227552. q^{31} +213360. q^{35} -160526. q^{37} -10842.0 q^{41} -630748. q^{43} +472656. q^{47} +208713. q^{49} -1.49402e6 q^{53} +229320. q^{55} -2.64066e6 q^{59} -827702. q^{61} +290220. q^{65} -126004. q^{67} -1.41473e6 q^{71} +980282. q^{73} +1.10947e6 q^{77} +3.56680e6 q^{79} -5.67289e6 q^{83} +3.08826e6 q^{85} +1.19512e7 q^{89} +1.40411e6 q^{91} +8.38740e6 q^{95} +8.68215e6 q^{97} +O(q^{100})$

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
$5$	−210.000	−0.751319				−0.375659	−	0.926758i	$-0.622584\pi$
			−0.375659	+	0.926758i	$0.622584\pi$
$7$	−1016.00	−1.11957	−0.559784	−	0.828638i	$-0.689116\pi$
			−0.559784	+	0.828638i	$0.689116\pi$
$11$	−1092.00	−0.247371	−0.123685	−	0.992321i	$-0.539471\pi$
			−0.123685	+	0.992321i	$0.539471\pi$
$13$	−1382.00	−0.174464	−0.0872321	−	0.996188i	$-0.527802\pi$
			−0.0872321	+	0.996188i	$0.527802\pi$
$17$	−14706.0	−0.725978	−0.362989	−	0.931793i	$-0.618244\pi$
			−0.362989	+	0.931793i	$0.618244\pi$
$19$	−39940.0	−1.33589	−0.667945	−	0.744211i	$-0.732826\pi$
			−0.667945	+	0.744211i	$0.732826\pi$
$23$	68712.0	1.17757	0.588783	−	0.808291i	$-0.299607\pi$
			0.588783	+	0.808291i	$0.299607\pi$
$29$	−102570.	−0.780957	−0.390479	−	0.920612i	$-0.627690\pi$
			−0.390479	+	0.920612i	$0.627690\pi$
$31$	−227552.	−1.37188	−0.685938	−	0.727660i	$-0.740608\pi$
			−0.685938	+	0.727660i	$0.740608\pi$
$37$	−160526.	−0.521002	−0.260501	−	0.965474i	$-0.583888\pi$
			−0.260501	+	0.965474i	$0.583888\pi$
$41$	−10842.0	−0.0245678	−0.0122839	−	0.999925i	$-0.503910\pi$
			−0.0122839	+	0.999925i	$0.503910\pi$
$43$	−630748.	−1.20981	−0.604904	−	0.796299i	$-0.706788\pi$
			−0.604904	+	0.796299i	$0.706788\pi$
$47$	472656.	0.664053	0.332026	−	0.943270i	$-0.392268\pi$
			0.332026	+	0.943270i	$0.392268\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.8.a.f.1.1		1
3.2	odd	2		64.8.a.e.1.1		1
4.3	odd	2		576.8.a.g.1.1		1
8.3	odd	2		18.8.a.b.1.1		1
8.5	even	2		144.8.a.i.1.1		1
12.11	even	2		64.8.a.c.1.1		1
24.5	odd	2		16.8.a.b.1.1		1
24.11	even	2		2.8.a.a.1.1	✓	1
40.3	even	4		450.8.c.g.199.1		2
40.19	odd	2		450.8.a.c.1.1		1
40.27	even	4		450.8.c.g.199.2		2
48.5	odd	4		256.8.b.f.129.1		2
48.11	even	4		256.8.b.b.129.2		2
48.29	odd	4		256.8.b.f.129.2		2
48.35	even	4		256.8.b.b.129.1		2
72.11	even	6		162.8.c.l.109.1		2
72.43	odd	6		162.8.c.a.109.1		2
72.59	even	6		162.8.c.l.55.1		2
72.67	odd	6		162.8.c.a.55.1		2
120.29	odd	2		400.8.a.l.1.1		1
120.53	even	4		400.8.c.j.49.1		2
120.59	even	2		50.8.a.g.1.1		1
120.77	even	4		400.8.c.j.49.2		2
120.83	odd	4		50.8.b.c.49.2		2
120.107	odd	4		50.8.b.c.49.1		2
168.11	even	6		98.8.c.d.79.1		2
168.59	odd	6		98.8.c.e.79.1		2
168.83	odd	2		98.8.a.a.1.1		1
168.107	even	6		98.8.c.d.67.1		2
168.131	odd	6		98.8.c.e.67.1		2
264.131	odd	2		242.8.a.e.1.1		1
312.83	odd	4		338.8.b.d.337.2		2
312.155	even	2		338.8.a.d.1.1		1
312.203	odd	4		338.8.b.d.337.1		2
408.203	even	2		578.8.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	24.11	even	2
16.8.a.b.1.1		1	24.5	odd	2
18.8.a.b.1.1		1	8.3	odd	2
50.8.a.g.1.1		1	120.59	even	2
50.8.b.c.49.1		2	120.107	odd	4
50.8.b.c.49.2		2	120.83	odd	4
64.8.a.c.1.1		1	12.11	even	2
64.8.a.e.1.1		1	3.2	odd	2
98.8.a.a.1.1		1	168.83	odd	2
98.8.c.d.67.1		2	168.107	even	6
98.8.c.d.79.1		2	168.11	even	6
98.8.c.e.67.1		2	168.131	odd	6
98.8.c.e.79.1		2	168.59	odd	6
144.8.a.i.1.1		1	8.5	even	2
162.8.c.a.55.1		2	72.67	odd	6
162.8.c.a.109.1		2	72.43	odd	6
162.8.c.l.55.1		2	72.59	even	6
162.8.c.l.109.1		2	72.11	even	6
242.8.a.e.1.1		1	264.131	odd	2
256.8.b.b.129.1		2	48.35	even	4
256.8.b.b.129.2		2	48.11	even	4
256.8.b.f.129.1		2	48.5	odd	4
256.8.b.f.129.2		2	48.29	odd	4
338.8.a.d.1.1		1	312.155	even	2
338.8.b.d.337.1		2	312.203	odd	4
338.8.b.d.337.2		2	312.83	odd	4
400.8.a.l.1.1		1	120.29	odd	2
400.8.c.j.49.1		2	120.53	even	4
400.8.c.j.49.2		2	120.77	even	4
450.8.a.c.1.1		1	40.19	odd	2
450.8.c.g.199.1		2	40.3	even	4
450.8.c.g.199.2		2	40.27	even	4
576.8.a.f.1.1		1	1.1	even	1	trivial
576.8.a.g.1.1		1	4.3	odd	2
578.8.a.b.1.1		1	408.203	even	2