Embedded newform 96.3.p.a.5.19

• Label: 96.3.p.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.576838 + 1.91501i) q^{2} +(-1.55085 - 2.56805i) q^{3} +(-3.33452 + 2.20930i) q^{4} +(3.63719 - 8.78095i) q^{5} +(4.02324 - 4.45124i) q^{6} +(-5.54083 - 5.54083i) q^{7} +(-6.15430 - 5.11122i) q^{8} +(-4.18972 + 7.96531i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.576838	+	1.91501i	0.288419	+	0.957504i
							\(3\)	−1.55085	−	2.56805i	−0.516950	−	0.856015i
\(5\)	3.63719	−	8.78095i	0.727438	−	1.75619i								0.0764857	−	0.997071i	\(-0.475630\pi\)
							0.650952	−	0.759119i	\(-0.274370\pi\)
\(7\)	−5.54083	−	5.54083i	−0.791547	−	0.791547i	0.190198	−	0.981746i	\(-0.439087\pi\)
							−0.981746	+	0.190198i	\(0.939087\pi\)
\(11\)	7.10294	+	2.94213i	0.645722	+	0.267467i	0.681416	−	0.731896i	\(-0.261364\pi\)
							−0.0356945	+	0.999363i	\(0.511364\pi\)
\(13\)	8.73200	−	3.61691i	0.671692	−	0.278224i	−0.0206571	−	0.999787i	\(-0.506576\pi\)
							0.692349	+	0.721563i	\(0.256576\pi\)
\(17\)	14.7964			0.870375			0.435188	−	0.900340i	\(-0.356682\pi\)
							0.435188	+	0.900340i	\(0.356682\pi\)
\(19\)	−18.9797	+	7.86164i	−0.998930	+	0.413770i	−0.821405	−	0.570346i	\(-0.806809\pi\)
							−0.177525	+	0.984116i	\(0.556809\pi\)
\(23\)	1.32792	+	1.32792i	0.0577358	+	0.0577358i	0.735385	−	0.677649i	\(-0.237001\pi\)
							−0.677649	+	0.735385i	\(0.737001\pi\)
\(29\)	3.54740	−	1.46938i	0.122324	−	0.0506683i	−0.320682	−	0.947187i	\(-0.603912\pi\)
							0.443006	+	0.896519i	\(0.353912\pi\)
\(31\)	−2.24816			−0.0725214			−0.0362607	−	0.999342i	\(-0.511545\pi\)
							−0.0362607	+	0.999342i	\(0.511545\pi\)
\(37\)	43.6786	+	18.0923i	1.18050	+	0.488980i	0.884651	−	0.466253i	\(-0.154396\pi\)
							0.295852	+	0.955234i	\(0.404396\pi\)
\(41\)	26.1721	+	26.1721i	0.638344	+	0.638344i	0.950147	−	0.311803i	\(-0.100933\pi\)
							−0.311803	+	0.950147i	\(0.600933\pi\)
\(43\)	7.76495	−	18.7462i	0.180580	−	0.435959i	−0.807506	−	0.589859i	\(-0.799183\pi\)
							0.988086	+	0.153900i	\(0.0491833\pi\)
\(47\)	36.7696			0.782333			0.391166	−	0.920320i	\(-0.372072\pi\)
							0.391166	+	0.920320i	\(0.372072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.19	yes	120
3.2	odd	2	inner	96.3.p.a.5.12	✓	120
4.3	odd	2		384.3.p.a.113.21		120
12.11	even	2		384.3.p.a.113.18		120
32.13	even	8	inner	96.3.p.a.77.12	yes	120
32.19	odd	8		384.3.p.a.17.18		120
96.77	odd	8	inner	96.3.p.a.77.19	yes	120
96.83	even	8		384.3.p.a.17.21		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.12	✓	120	3.2	odd	2	inner
96.3.p.a.5.19	yes	120	1.1	even	1	trivial
96.3.p.a.77.12	yes	120	32.13	even	8	inner
96.3.p.a.77.19	yes	120	96.77	odd	8	inner
384.3.p.a.17.18		120	32.19	odd	8
384.3.p.a.17.21		120	96.83	even	8
384.3.p.a.113.18		120	12.11	even	2
384.3.p.a.113.21		120	4.3	odd	2