Embedded newform 96.3.p.a.5.4

• Label: 96.3.p.a.5.4
• Level: $96$
• Weight: $3$
• Character: 96.5
• Analytic conductor: $2.616$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.61581053786\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84984 - 0.760319i) q^{2} +(-0.317768 + 2.98312i) q^{3} +(2.84383 + 2.81294i) q^{4} +(-1.77368 + 4.28204i) q^{5} +(2.85595 - 5.27670i) q^{6} +(-7.55140 - 7.55140i) q^{7} +(-3.12190 - 7.36571i) q^{8} +(-8.79805 - 1.89588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84984	−	0.760319i	−0.924921	−	0.380160i
							\(3\)	−0.317768	+	2.98312i	−0.105923	+	0.994374i
\(5\)	−1.77368	+	4.28204i	−0.354736	+	0.856408i								0.641286	+	0.767302i	\(0.278401\pi\)
							−0.996022	+	0.0891063i	\(0.971599\pi\)
\(7\)	−7.55140	−	7.55140i	−1.07877	−	1.07877i	−0.996620	−	0.0821517i	\(-0.973821\pi\)
							−0.0821517	−	0.996620i	\(-0.526179\pi\)
\(11\)	8.31387	+	3.44372i	0.755807	+	0.313065i	0.727108	−	0.686523i	\(-0.240864\pi\)
							0.0286983	+	0.999588i	\(0.490864\pi\)
\(13\)	−18.9971	+	7.86886i	−1.46132	+	0.605297i	−0.964860	−	0.262764i	\(-0.915366\pi\)
							−0.496457	+	0.868061i	\(0.665366\pi\)
\(17\)	−17.6974			−1.04102			−0.520512	−	0.853855i	\(-0.674259\pi\)
							−0.520512	+	0.853855i	\(0.674259\pi\)
\(19\)	8.02832	−	3.32544i	0.422543	−	0.175023i	−0.161272	−	0.986910i	\(-0.551559\pi\)
							0.583815	+	0.811887i	\(0.301559\pi\)
\(23\)	12.1912	+	12.1912i	0.530053	+	0.530053i	0.920588	−	0.390535i	\(-0.127710\pi\)
							−0.390535	+	0.920588i	\(0.627710\pi\)
\(29\)	−38.1346	+	15.7959i	−1.31499	+	0.544685i	−0.926335	−	0.376700i	\(-0.877059\pi\)
							−0.388651	+	0.921385i	\(0.627059\pi\)
\(31\)	−15.3070			−0.493776			−0.246888	−	0.969044i	\(-0.579408\pi\)
							−0.246888	+	0.969044i	\(0.579408\pi\)
\(37\)	43.6037	+	18.0612i	1.17848	+	0.488142i	0.883987	−	0.467512i	\(-0.154850\pi\)
							0.294492	+	0.955654i	\(0.404850\pi\)
\(41\)	40.8848	+	40.8848i	0.997191	+	0.997191i	0.999996	−	0.00280499i	\(-0.000892857\pi\)
							−0.00280499	+	0.999996i	\(0.500893\pi\)
\(43\)	−5.11152	+	12.3403i	−0.118872	+	0.286983i	−0.972105	−	0.234548i	\(-0.924639\pi\)
							0.853232	+	0.521531i	\(0.174639\pi\)
\(47\)	−17.9203			−0.381284			−0.190642	−	0.981660i	\(-0.561057\pi\)
							−0.190642	+	0.981660i	\(0.561057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.4	✓	120
3.2	odd	2	inner	96.3.p.a.5.27	yes	120
4.3	odd	2		384.3.p.a.113.16		120
12.11	even	2		384.3.p.a.113.6		120
32.13	even	8	inner	96.3.p.a.77.27	yes	120
32.19	odd	8		384.3.p.a.17.6		120
96.77	odd	8	inner	96.3.p.a.77.4	yes	120
96.83	even	8		384.3.p.a.17.16		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.4	✓	120	1.1	even	1	trivial
96.3.p.a.5.27	yes	120	3.2	odd	2	inner
96.3.p.a.77.4	yes	120	96.77	odd	8	inner
96.3.p.a.77.27	yes	120	32.13	even	8	inner
384.3.p.a.17.6		120	32.19	odd	8
384.3.p.a.17.16		120	96.83	even	8
384.3.p.a.113.6		120	12.11	even	2
384.3.p.a.113.16		120	4.3	odd	2