Embedded newform 96.3.p.a.5.18

• Label: 96.3.p.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.503078 - 1.93569i) q^{2} +(-2.39365 + 1.80844i) q^{3} +(-3.49383 - 1.94761i) q^{4} +(-2.54567 + 6.14580i) q^{5} +(2.29639 + 5.54316i) q^{6} +(7.32134 + 7.32134i) q^{7} +(-5.52764 + 5.78318i) q^{8} +(2.45911 - 8.65753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.503078	−	1.93569i	0.251539	−	0.967847i
							\(3\)	−2.39365	+	1.80844i	−0.797883	+	0.602812i
\(5\)	−2.54567	+	6.14580i	−0.509135	+	1.22916i								0.435248	+	0.900311i	\(0.356661\pi\)
							−0.944382	+	0.328849i	\(0.893339\pi\)
\(7\)	7.32134	+	7.32134i	1.04591	+	1.04591i	0.998894	+	0.0470115i	\(0.0149697\pi\)
							0.0470115	+	0.998894i	\(0.485030\pi\)
\(11\)	3.73475	+	1.54698i	0.339522	+	0.140635i	0.545929	−	0.837832i	\(-0.316177\pi\)
							−0.206406	+	0.978466i	\(0.566177\pi\)
\(13\)	−20.0259	+	8.29499i	−1.54045	+	0.638076i	−0.981558	−	0.191163i	\(-0.938774\pi\)
							−0.558894	+	0.829239i	\(0.688774\pi\)
\(17\)	−11.4146			−0.671447			−0.335724	−	0.941961i	\(-0.608981\pi\)
							−0.335724	+	0.941961i	\(0.608981\pi\)
\(19\)	−2.41656	+	1.00097i	−0.127187	+	0.0526828i	−0.445369	−	0.895347i	\(-0.646928\pi\)
							0.318182	+	0.948030i	\(0.396928\pi\)
\(23\)	10.6804	+	10.6804i	0.464367	+	0.464367i	0.900084	−	0.435717i	\(-0.143505\pi\)
							−0.435717	+	0.900084i	\(0.643505\pi\)
\(29\)	38.4375	−	15.9213i	1.32543	−	0.549012i	0.396083	−	0.918215i	\(-0.370369\pi\)
							0.929349	+	0.369203i	\(0.120369\pi\)
\(31\)	22.9544			0.740466			0.370233	−	0.928939i	\(-0.379278\pi\)
							0.370233	+	0.928939i	\(0.379278\pi\)
\(37\)	−47.4669	−	19.6614i	−1.28289	−	0.531390i	−0.366031	−	0.930603i	\(-0.619284\pi\)
							−0.916858	+	0.399212i	\(0.869284\pi\)
\(41\)	−7.45051	−	7.45051i	−0.181720	−	0.181720i	0.610385	−	0.792105i	\(-0.291015\pi\)
							−0.792105	+	0.610385i	\(0.791015\pi\)
\(43\)	11.2795	−	27.2311i	0.262314	−	0.633281i	−0.736767	−	0.676146i	\(-0.763649\pi\)
							0.999081	+	0.0428655i	\(0.0136487\pi\)
\(47\)	15.6846			0.333715			0.166857	−	0.985981i	\(-0.446638\pi\)
							0.166857	+	0.985981i	\(0.446638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.18	yes	120
3.2	odd	2	inner	96.3.p.a.5.13	✓	120
4.3	odd	2		384.3.p.a.113.25		120
12.11	even	2		384.3.p.a.113.2		120
32.13	even	8	inner	96.3.p.a.77.13	yes	120
32.19	odd	8		384.3.p.a.17.2		120
96.77	odd	8	inner	96.3.p.a.77.18	yes	120
96.83	even	8		384.3.p.a.17.25		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.13	✓	120	3.2	odd	2	inner
96.3.p.a.5.18	yes	120	1.1	even	1	trivial
96.3.p.a.77.13	yes	120	32.13	even	8	inner
96.3.p.a.77.18	yes	120	96.77	odd	8	inner
384.3.p.a.17.2		120	32.19	odd	8
384.3.p.a.17.25		120	96.83	even	8
384.3.p.a.113.2		120	12.11	even	2
384.3.p.a.113.25		120	4.3	odd	2