Embedded newform 96.3.m.a.19.4

• Label: 96.3.m.a.19.4
• 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			19.4
Character	\(\chi\)	\(=\)	96.19
Dual form			96.3.m.a.91.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49565 - 1.32779i) q^{2} +(-1.60021 - 0.662827i) q^{3} +(0.473939 + 3.97182i) q^{4} +(5.66092 - 2.34483i) q^{5} +(1.51325 + 3.11610i) q^{6} +(-8.77775 - 8.77775i) q^{7} +(4.56491 - 6.56975i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49565	−	1.32779i	−0.747825	−	0.663896i
							\(3\)	−1.60021	−	0.662827i	−0.533402	−	0.220942i
\(5\)	5.66092	−	2.34483i	1.13218	−	0.468966i								0.263661	−	0.964615i	\(-0.415070\pi\)
							0.868523	+	0.495650i	\(0.165070\pi\)
\(7\)	−8.77775	−	8.77775i	−1.25396	−	1.25396i	−0.953928	−	0.300036i	\(-0.903001\pi\)
							−0.300036	−	0.953928i	\(-0.596999\pi\)
\(11\)	−8.07224	+	3.34363i	−0.733840	+	0.303967i	−0.718129	−	0.695910i	\(-0.755001\pi\)
							−0.0157113	+	0.999877i	\(0.505001\pi\)
\(13\)	−14.6204	−	6.05596i	−1.12465	−	0.465843i	−0.258688	−	0.965961i	\(-0.583290\pi\)
							−0.865957	+	0.500118i	\(0.833290\pi\)
\(17\)		−	5.04544i		−	0.296790i	−0.988928	−	0.148395i	\(-0.952589\pi\)
							0.988928	−	0.148395i	\(-0.0474108\pi\)
\(19\)	7.56553	−	18.2648i	0.398186	−	0.961306i	−0.589910	−	0.807469i	\(-0.700837\pi\)
							0.988096	−	0.153837i	\(-0.0491631\pi\)
\(23\)	15.3825	−	15.3825i	0.668804	−	0.668804i	−0.288635	−	0.957439i	\(-0.593201\pi\)
							0.957439	+	0.288635i	\(0.0932014\pi\)
\(29\)	13.8865	−	33.5250i	0.478846	−	1.15604i	−0.481305	−	0.876553i	\(-0.659837\pi\)
							0.960151	−	0.279482i	\(-0.0901629\pi\)
\(31\)			40.4409i			1.30454i	0.757985	+	0.652272i	\(0.226184\pi\)
							−0.757985	+	0.652272i	\(0.773816\pi\)
\(37\)	4.91805	−	2.03712i	0.132920	−	0.0550574i	−0.315232	−	0.949015i	\(-0.602082\pi\)
							0.448152	+	0.893957i	\(0.352082\pi\)
\(41\)	38.1935	+	38.1935i	0.931550	+	0.931550i	0.997803	−	0.0662532i	\(-0.0211045\pi\)
							−0.0662532	+	0.997803i	\(0.521105\pi\)
\(43\)	−15.3872	+	6.37357i	−0.357841	+	0.148223i	−0.554359	−	0.832278i	\(-0.687037\pi\)
							0.196518	+	0.980500i	\(0.437037\pi\)
\(47\)	64.7817			1.37833			0.689167	−	0.724603i	\(-0.257977\pi\)
							0.689167	+	0.724603i	\(0.257977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.m.a.19.4	✓	64
3.2	odd	2		288.3.u.b.19.13		64
4.3	odd	2		384.3.m.a.367.15		64
32.5	even	8		384.3.m.a.271.15		64
32.27	odd	8	inner	96.3.m.a.91.4	yes	64
96.59	even	8		288.3.u.b.91.13		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.m.a.19.4	✓	64	1.1	even	1	trivial
96.3.m.a.91.4	yes	64	32.27	odd	8	inner
288.3.u.b.19.13		64	3.2	odd	2
288.3.u.b.91.13		64	96.59	even	8
384.3.m.a.271.15		64	32.5	even	8
384.3.m.a.367.15		64	4.3	odd	2