Embedded newform 384.2.w.a.227.5

• Label: 384.2.w.a.227.5
• Level: $384$
• Weight: $2$
• Character: 384.227
• 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			227.5
Character	\(\chi\)	\(=\)	384.227
Dual form			384.2.w.a.203.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38570 + 0.282536i) q^{2} +(0.297411 + 1.70633i) q^{3} +(1.84035 - 0.783022i) q^{4} +(-1.06178 - 1.98646i) q^{5} +(-0.894221 - 2.28043i) q^{6} +(1.02659 - 0.204202i) q^{7} +(-2.32894 + 1.60500i) q^{8} +(-2.82309 + 1.01496i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38570	+	0.282536i	−0.979840	+	0.199783i
							\(3\)	0.297411	+	1.70633i	0.171710	+	0.985148i
\(5\)	−1.06178	−	1.98646i	−0.474845	−	0.888372i								−0.999317	−	0.0369444i	\(-0.988238\pi\)
							0.524473	−	0.851427i	\(-0.324262\pi\)
\(7\)	1.02659	−	0.204202i	0.388015	−	0.0771809i	0.00277194	−	0.999996i	\(-0.499118\pi\)
							0.385243	+	0.922815i	\(0.374118\pi\)
\(11\)	3.73656	−	4.55301i	1.12661	−	1.37278i	0.207763	−	0.978179i	\(-0.433382\pi\)
							0.918851	−	0.394604i	\(-0.129118\pi\)
\(13\)	1.37071	−	2.56441i	0.380166	−	0.711241i	−0.616961	−	0.786994i	\(-0.711636\pi\)
							0.997127	+	0.0757535i	\(0.0241362\pi\)
\(17\)	−0.811719	−	0.336225i	−0.196871	−	0.0815466i	0.282070	−	0.959394i	\(-0.408979\pi\)
							−0.478941	+	0.877847i	\(0.658979\pi\)
\(19\)	−3.88118	+	1.17734i	−0.890404	+	0.270101i	−0.702158	−	0.712021i	\(-0.747780\pi\)
							−0.188246	+	0.982122i	\(0.560280\pi\)
\(23\)	5.89204	−	3.93693i	1.22857	−	0.820908i	0.239873	−	0.970804i	\(-0.422894\pi\)
							0.988702	+	0.149897i	\(0.0478941\pi\)
\(29\)	4.65152	+	5.66790i	0.863766	+	1.05250i	0.998103	+	0.0615664i	\(0.0196096\pi\)
							−0.134337	+	0.990936i	\(0.542890\pi\)
\(31\)	−1.43245	−	1.43245i	−0.257275	−	0.257275i	0.566670	−	0.823945i	\(-0.308232\pi\)
							−0.823945	+	0.566670i	\(0.808232\pi\)
\(37\)	4.85815	+	1.47370i	0.798675	+	0.242275i	0.663138	−	0.748497i	\(-0.269224\pi\)
							0.135537	+	0.990772i	\(0.456724\pi\)
\(41\)	−5.92688	+	3.96021i	−0.925623	+	0.618481i	−0.924361	−	0.381518i	\(-0.875401\pi\)
							−0.00126114	+	0.999999i	\(0.500401\pi\)
\(43\)	8.34816	−	0.822222i	1.27308	−	0.125388i	0.561196	−	0.827683i	\(-0.310341\pi\)
							0.711886	+	0.702295i	\(0.247841\pi\)
\(47\)	5.52710	+	2.28940i	0.806211	+	0.333944i	0.747441	−	0.664328i	\(-0.231282\pi\)
							0.0587699	+	0.998272i	\(0.481282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.227.5	yes	992
3.2	odd	2	inner	384.2.w.a.227.58	yes	992
128.75	odd	32	inner	384.2.w.a.203.58	yes	992
384.203	even	32	inner	384.2.w.a.203.5	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.203.5	✓	992	384.203	even	32	inner
384.2.w.a.203.58	yes	992	128.75	odd	32	inner
384.2.w.a.227.5	yes	992	1.1	even	1	trivial
384.2.w.a.227.58	yes	992	3.2	odd	2	inner