Embedded newform 384.2.w.a.35.8

• Label: 384.2.w.a.35.8
• Level: $384$
• Weight: $2$
• Character: 384.35
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.w (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06625543762\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			35.8
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29156 + 0.576082i) q^{2} +(-1.73162 - 0.0387632i) q^{3} +(1.33626 - 1.48809i) q^{4} +(-0.861594 + 0.460532i) q^{5} +(2.25882 - 0.947488i) q^{6} +(-4.16278 + 0.828028i) q^{7} +(-0.868599 + 2.69175i) q^{8} +(2.99699 + 0.134246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29156	+	0.576082i	−0.913272	+	0.407351i
							\(3\)	−1.73162	−	0.0387632i	−0.999750	−	0.0223800i
\(5\)	−0.861594	+	0.460532i	−0.385317	+	0.205956i								−0.652670	−	0.757643i	\(-0.726351\pi\)
							0.267353	+	0.963599i	\(0.413851\pi\)
\(7\)	−4.16278	+	0.828028i	−1.57338	+	0.312965i	−0.903196	−	0.429229i	\(-0.858785\pi\)
							−0.670185	+	0.742194i	\(0.733785\pi\)
\(11\)	0.0821496	+	0.0674185i	0.0247691	+	0.0203274i	0.646696	−	0.762748i	\(-0.276150\pi\)
							−0.621927	+	0.783075i	\(0.713650\pi\)
\(13\)	2.83393	+	1.51477i	0.785992	+	0.420122i	0.815002	−	0.579458i	\(-0.196736\pi\)
							−0.0290103	+	0.999579i	\(0.509236\pi\)
\(17\)	−1.91264	−	0.792239i	−0.463882	−	0.192146i	0.138486	−	0.990364i	\(-0.455776\pi\)
							−0.602369	+	0.798218i	\(0.705776\pi\)
\(19\)	−1.97609	−	6.51431i	−0.453347	−	1.49448i	−0.824473	−	0.565902i	\(-0.808528\pi\)
							0.371126	−	0.928583i	\(-0.378972\pi\)
\(23\)	5.87484	−	3.92544i	1.22499	−	0.818512i	0.236771	−	0.971565i	\(-0.423911\pi\)
							0.988218	+	0.153054i	\(0.0489108\pi\)
\(29\)	8.21329	−	6.74047i	1.52517	−	1.25167i	0.658743	−	0.752368i	\(-0.271089\pi\)
							0.866426	−	0.499305i	\(-0.166411\pi\)
\(31\)	5.10979	+	5.10979i	0.917745	+	0.917745i	0.996865	−	0.0791202i	\(-0.0252111\pi\)
							−0.0791202	+	0.996865i	\(0.525211\pi\)
\(37\)	1.79427	−	5.91491i	0.294976	−	0.972406i	−0.676734	−	0.736228i	\(-0.736605\pi\)
							0.971710	−	0.236178i	\(-0.0758948\pi\)
\(41\)	2.56549	−	1.71420i	0.400662	−	0.267714i	−0.338873	−	0.940832i	\(-0.610046\pi\)
							0.739535	+	0.673119i	\(0.235046\pi\)
\(43\)	−0.403187	−	4.09362i	−0.0614854	−	0.624271i	−0.975710	−	0.219065i	\(-0.929699\pi\)
							0.914225	−	0.405207i	\(-0.132801\pi\)
\(47\)	9.29317	+	3.84936i	1.35555	+	0.561486i	0.937831	−	0.347091i	\(-0.112831\pi\)
							0.417716	+	0.908578i	\(0.362831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.35.8	yes	992
3.2	odd	2	inner	384.2.w.a.35.55	yes	992
128.11	odd	32	inner	384.2.w.a.11.55	yes	992
384.11	even	32	inner	384.2.w.a.11.8	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.8	✓	992	384.11	even	32	inner
384.2.w.a.11.55	yes	992	128.11	odd	32	inner
384.2.w.a.35.8	yes	992	1.1	even	1	trivial
384.2.w.a.35.55	yes	992	3.2	odd	2	inner