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