Embedded newform 96.3.m.a.19.5

• Label: 96.3.m.a.19.5
• Level: $96$
• Weight: $3$
• Character: 96.19
• Analytic conductor: $2.616$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.61581053786\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			19.5
Character	\(\chi\)	\(=\)	96.19
Dual form			96.3.m.a.91.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46158 + 1.36521i) q^{2} +(1.60021 + 0.662827i) q^{3} +(0.272417 - 3.99071i) q^{4} +(-1.99171 + 0.824993i) q^{5} +(-3.24372 + 1.21584i) q^{6} +(8.28172 + 8.28172i) q^{7} +(5.04999 + 6.20464i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46158	+	1.36521i	−0.730789	+	0.682604i
							\(3\)	1.60021	+	0.662827i	0.533402	+	0.220942i
\(5\)	−1.99171	+	0.824993i	−0.398342	+	0.164999i								−0.572856	−	0.819656i	\(-0.694165\pi\)
							0.174514	+	0.984655i	\(0.444165\pi\)
\(7\)	8.28172	+	8.28172i	1.18310	+	1.18310i	0.978936	+	0.204167i	\(0.0654485\pi\)
							0.204167	+	0.978936i	\(0.434551\pi\)
\(11\)	−8.98118	+	3.72013i	−0.816471	+	0.338193i	−0.751532	−	0.659696i	\(-0.770685\pi\)
							−0.0649381	+	0.997889i	\(0.520685\pi\)
\(13\)	5.35333	+	2.21742i	0.411794	+	0.170571i	0.578956	−	0.815359i	\(-0.303460\pi\)
							−0.167162	+	0.985929i	\(0.553460\pi\)
\(17\)			23.6791i			1.39289i	0.717610	+	0.696445i	\(0.245236\pi\)
							−0.717610	+	0.696445i	\(0.754764\pi\)
\(19\)	5.00496	−	12.0830i	0.263419	−	0.635950i	−0.735727	−	0.677279i	\(-0.763159\pi\)
							0.999146	+	0.0413288i	\(0.0131591\pi\)
\(23\)	19.0235	−	19.0235i	0.827109	−	0.827109i	−0.160007	−	0.987116i	\(-0.551152\pi\)
							0.987116	+	0.160007i	\(0.0511517\pi\)
\(29\)	11.9161	−	28.7681i	0.410902	−	0.992004i	−0.573995	−	0.818859i	\(-0.694607\pi\)
							0.984896	−	0.173145i	\(-0.0553930\pi\)
\(31\)		−	47.9498i		−	1.54677i	−0.633938	−	0.773384i	\(-0.718563\pi\)
							0.633938	−	0.773384i	\(-0.281437\pi\)
\(37\)	10.5107	−	4.35369i	0.284074	−	0.117667i	−0.236097	−	0.971730i	\(-0.575868\pi\)
							0.520171	+	0.854062i	\(0.325868\pi\)
\(41\)	5.41009	+	5.41009i	0.131953	+	0.131953i	0.769999	−	0.638045i	\(-0.220257\pi\)
							−0.638045	+	0.769999i	\(0.720257\pi\)
\(43\)	34.1273	−	14.1360i	0.793658	−	0.328744i	0.0512445	−	0.998686i	\(-0.483681\pi\)
							0.742413	+	0.669942i	\(0.233681\pi\)
\(47\)	−9.65109			−0.205342			−0.102671	−	0.994715i	\(-0.532739\pi\)
							−0.102671	+	0.994715i	\(0.532739\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.m.a.19.5	✓	64
3.2	odd	2		288.3.u.b.19.12		64
4.3	odd	2		384.3.m.a.367.3		64
32.5	even	8		384.3.m.a.271.3		64
32.27	odd	8	inner	96.3.m.a.91.5	yes	64
96.59	even	8		288.3.u.b.91.12		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.m.a.19.5	✓	64	1.1	even	1	trivial
96.3.m.a.91.5	yes	64	32.27	odd	8	inner
288.3.u.b.19.12		64	3.2	odd	2
288.3.u.b.91.12		64	96.59	even	8
384.3.m.a.271.3		64	32.5	even	8
384.3.m.a.367.3		64	4.3	odd	2