Embedded newform 96.3.p.a.5.6

• Label: 96.3.p.a.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50790 - 1.31386i) q^{2} +(-2.99990 + 0.0247672i) q^{3} +(0.547547 + 3.96235i) q^{4} +(-0.0391005 + 0.0943970i) q^{5} +(4.55610 + 3.90410i) q^{6} +(2.42911 + 2.42911i) q^{7} +(4.38032 - 6.69424i) q^{8} +(8.99877 - 0.148598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50790	−	1.31386i	−0.753952	−	0.656930i
							\(3\)	−2.99990	+	0.0247672i	−0.999966	+	0.00825572i
\(5\)	−0.0391005	+	0.0943970i	−0.00782011	+	0.0188794i								−0.927741	−	0.373224i	\(-0.878252\pi\)
							0.919921	+	0.392103i	\(0.128252\pi\)
\(7\)	2.42911	+	2.42911i	0.347016	+	0.347016i	0.858997	−	0.511981i	\(-0.171088\pi\)
							−0.511981	+	0.858997i	\(0.671088\pi\)
\(11\)	5.15164	+	2.13388i	0.468331	+	0.193989i	0.604353	−	0.796717i	\(-0.293432\pi\)
							−0.136022	+	0.990706i	\(0.543432\pi\)
\(13\)	14.3035	−	5.92469i	1.10027	−	0.455746i	0.242692	−	0.970103i	\(-0.421970\pi\)
							0.857576	+	0.514358i	\(0.171970\pi\)
\(17\)	24.5126			1.44192			0.720958	−	0.692978i	\(-0.243702\pi\)
							0.720958	+	0.692978i	\(0.243702\pi\)
\(19\)	−18.0084	+	7.45931i	−0.947809	+	0.392595i	−0.802407	−	0.596778i	\(-0.796447\pi\)
							−0.145402	+	0.989373i	\(0.546447\pi\)
\(23\)	17.5944	+	17.5944i	0.764974	+	0.764974i	0.977217	−	0.212243i	\(-0.0680769\pi\)
							−0.212243	+	0.977217i	\(0.568077\pi\)
\(29\)	−14.2794	+	5.91471i	−0.492392	+	0.203956i	−0.615042	−	0.788494i	\(-0.710861\pi\)
							0.122649	+	0.992450i	\(0.460861\pi\)
\(31\)	−4.69346			−0.151402			−0.0757009	−	0.997131i	\(-0.524119\pi\)
							−0.0757009	+	0.997131i	\(0.524119\pi\)
\(37\)	15.1074	+	6.25769i	0.408308	+	0.169127i	0.577377	−	0.816477i	\(-0.304076\pi\)
							−0.169069	+	0.985604i	\(0.554076\pi\)
\(41\)	−8.09729	−	8.09729i	−0.197495	−	0.197495i	0.601430	−	0.798925i	\(-0.294598\pi\)
							−0.798925	+	0.601430i	\(0.794598\pi\)
\(43\)	14.8848	−	35.9350i	0.346158	−	0.835698i	−0.650909	−	0.759156i	\(-0.725612\pi\)
							0.997066	−	0.0765422i	\(-0.0243880\pi\)
\(47\)	26.9280			0.572935			0.286468	−	0.958090i	\(-0.407519\pi\)
							0.286468	+	0.958090i	\(0.407519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.6	✓	120
3.2	odd	2	inner	96.3.p.a.5.25	yes	120
4.3	odd	2		384.3.p.a.113.30		120
12.11	even	2		384.3.p.a.113.9		120
32.13	even	8	inner	96.3.p.a.77.25	yes	120
32.19	odd	8		384.3.p.a.17.9		120
96.77	odd	8	inner	96.3.p.a.77.6	yes	120
96.83	even	8		384.3.p.a.17.30		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.6	✓	120	1.1	even	1	trivial
96.3.p.a.5.25	yes	120	3.2	odd	2	inner
96.3.p.a.77.6	yes	120	96.77	odd	8	inner
96.3.p.a.77.25	yes	120	32.13	even	8	inner
384.3.p.a.17.9		120	32.19	odd	8
384.3.p.a.17.30		120	96.83	even	8
384.3.p.a.113.9		120	12.11	even	2
384.3.p.a.113.30		120	4.3	odd	2