Embedded newform 384.2.w.a.11.6

• Label: 384.2.w.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30952 + 0.533994i) q^{2} +(1.15242 + 1.29303i) q^{3} +(1.42970 - 1.39855i) q^{4} +(3.25884 + 1.74189i) q^{5} +(-2.19959 - 1.07787i) q^{6} +(-3.48323 - 0.692857i) q^{7} +(-1.12541 + 2.59489i) q^{8} +(-0.343871 + 2.98023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30952	+	0.533994i	−0.925973	+	0.377591i
							\(3\)	1.15242	+	1.29303i	0.665348	+	0.746533i
\(5\)	3.25884	+	1.74189i	1.45740	+	0.778995i								0.994229	−	0.107276i	\(-0.0342127\pi\)
							0.463168	+	0.886271i	\(0.346713\pi\)
\(7\)	−3.48323	−	0.692857i	−1.31654	−	0.261875i	−0.513646	−	0.858002i	\(-0.671705\pi\)
							−0.802890	+	0.596127i	\(0.796705\pi\)
\(11\)	−4.31500	+	3.54123i	−1.30102	+	1.06772i	−0.307804	+	0.951450i	\(0.599594\pi\)
							−0.993218	+	0.116271i	\(0.962906\pi\)
\(13\)	−1.49393	+	0.798522i	−0.414342	+	0.221470i	−0.665377	−	0.746508i	\(-0.731729\pi\)
							0.251035	+	0.967978i	\(0.419229\pi\)
\(17\)	4.64448	−	1.92380i	1.12645	−	0.466591i	0.259878	−	0.965642i	\(-0.416318\pi\)
							0.866573	+	0.499050i	\(0.166318\pi\)
\(19\)	−0.131854	+	0.434666i	−0.0302495	+	0.0997192i	−0.970765	−	0.240031i	\(-0.922842\pi\)
							0.940516	+	0.339750i	\(0.110342\pi\)
\(23\)	3.35062	+	2.23881i	0.698652	+	0.466824i	0.853486	−	0.521116i	\(-0.174484\pi\)
							−0.154834	+	0.987941i	\(0.549484\pi\)
\(29\)	−2.34537	−	1.92480i	−0.435525	−	0.357426i	0.390938	−	0.920417i	\(-0.372151\pi\)
							−0.826463	+	0.562991i	\(0.809651\pi\)
\(31\)	6.12586	−	6.12586i	1.10024	−	1.10024i	0.105856	−	0.994382i	\(-0.466242\pi\)
							0.994382	−	0.105856i	\(-0.0337582\pi\)
\(37\)	1.12796	+	3.71840i	0.185436	+	0.611300i	0.999533	+	0.0305710i	\(0.00973257\pi\)
							−0.814097	+	0.580729i	\(0.802767\pi\)
\(41\)	5.02931	+	3.36048i	0.785446	+	0.524818i	0.882417	−	0.470469i	\(-0.155915\pi\)
							−0.0969703	+	0.995287i	\(0.530915\pi\)
\(43\)	−0.0458460	+	0.465482i	−0.00699145	+	0.0709854i	−0.998047	−	0.0624621i	\(-0.980105\pi\)
							0.991056	+	0.133447i	\(0.0426047\pi\)
\(47\)	7.08876	−	2.93626i	1.03400	−	0.428298i	0.199847	−	0.979827i	\(-0.435956\pi\)
							0.834155	+	0.551530i	\(0.185956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.6	✓	992
3.2	odd	2	inner	384.2.w.a.11.57	yes	992
128.35	odd	32	inner	384.2.w.a.35.57	yes	992
384.35	even	32	inner	384.2.w.a.35.6	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.6	✓	992	1.1	even	1	trivial
384.2.w.a.11.57	yes	992	3.2	odd	2	inner
384.2.w.a.35.6	yes	992	384.35	even	32	inner
384.2.w.a.35.57	yes	992	128.35	odd	32	inner