Embedded newform 384.2.w.a.11.12

• Label: 384.2.w.a.11.12
• Level: $384$
• Weight: $2$
• Character: 384.11
• 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			11.12
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12499 + 0.856964i) q^{2} +(0.434523 + 1.67666i) q^{3} +(0.531225 - 1.92816i) q^{4} +(-2.13158 - 1.13935i) q^{5} +(-1.92567 - 1.51386i) q^{6} +(1.86189 + 0.370353i) q^{7} +(1.05474 + 2.62441i) q^{8} +(-2.62238 + 1.45710i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12499	+	0.856964i	−0.795491	+	0.605965i
							\(3\)	0.434523	+	1.67666i	0.250872	+	0.968020i
\(5\)	−2.13158	−	1.13935i	−0.953271	−	0.509534i								−0.0800138	−	0.996794i	\(-0.525496\pi\)
							−0.873257	+	0.487260i	\(0.837996\pi\)
\(7\)	1.86189	+	0.370353i	0.703729	+	0.139980i	0.533968	−	0.845505i	\(-0.320700\pi\)
							0.169761	+	0.985485i	\(0.445700\pi\)
\(11\)	−1.11565	+	0.915592i	−0.336382	+	0.276061i	−0.787386	−	0.616461i	\(-0.788566\pi\)
							0.451004	+	0.892522i	\(0.351066\pi\)
\(13\)	−3.82121	+	2.04248i	−1.05981	+	0.566482i	−0.906721	−	0.421730i	\(-0.861423\pi\)
							−0.153091	+	0.988212i	\(0.548923\pi\)
\(17\)	−4.29024	+	1.77708i	−1.04054	+	0.431004i	−0.836505	−	0.547960i	\(-0.815405\pi\)
							−0.204032	+	0.978964i	\(0.565405\pi\)
\(19\)	−1.29196	+	4.25902i	−0.296396	+	0.977087i	0.674638	+	0.738148i	\(0.264300\pi\)
							−0.971035	+	0.238939i	\(0.923200\pi\)
\(23\)	−5.93323	−	3.96446i	−1.23716	−	0.826647i	−0.247339	−	0.968929i	\(-0.579556\pi\)
							−0.989826	+	0.142282i	\(0.954556\pi\)
\(29\)	7.57068	+	6.21309i	1.40584	+	1.15374i	0.966879	+	0.255236i	\(0.0821532\pi\)
							0.438961	+	0.898506i	\(0.355347\pi\)
\(31\)	2.40772	−	2.40772i	0.432439	−	0.432439i	−0.457018	−	0.889457i	\(-0.651083\pi\)
							0.889457	+	0.457018i	\(0.151083\pi\)
\(37\)	−2.34496	−	7.73030i	−0.385509	−	1.27085i	−0.908274	−	0.418376i	\(-0.862599\pi\)
							0.522765	−	0.852477i	\(-0.324901\pi\)
\(41\)	3.34719	+	2.23652i	0.522743	+	0.349286i	0.788778	−	0.614678i	\(-0.210714\pi\)
							−0.266036	+	0.963963i	\(0.585714\pi\)
\(43\)	−0.764000	+	7.75702i	−0.116509	+	1.18293i	0.742095	+	0.670295i	\(0.233832\pi\)
							−0.858604	+	0.512640i	\(0.828668\pi\)
\(47\)	1.44341	−	0.597878i	0.210542	−	0.0872095i	−0.274920	−	0.961467i	\(-0.588651\pi\)
							0.485462	+	0.874258i	\(0.338651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.12	✓	992
3.2	odd	2	inner	384.2.w.a.11.51	yes	992
128.35	odd	32	inner	384.2.w.a.35.51	yes	992
384.35	even	32	inner	384.2.w.a.35.12	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.12	✓	992	1.1	even	1	trivial
384.2.w.a.11.51	yes	992	3.2	odd	2	inner
384.2.w.a.35.12	yes	992	384.35	even	32	inner
384.2.w.a.35.51	yes	992	128.35	odd	32	inner