Embedded newform 384.2.w.a.11.49

• Label: 384.2.w.a.11.49
• 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			11.49
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.11401 + 0.871200i) q^{2} +(-0.264236 + 1.71178i) q^{3} +(0.482021 + 1.94104i) q^{4} +(-3.29266 - 1.75996i) q^{5} +(-1.78566 + 1.67673i) q^{6} +(-0.370976 - 0.0737916i) q^{7} +(-1.15406 + 2.58227i) q^{8} +(-2.86036 - 0.904627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.11401	+	0.871200i	0.787722	+	0.616031i
							\(3\)	−0.264236	+	1.71178i	−0.152557	+	0.988295i
\(5\)	−3.29266	−	1.75996i	−1.47252	−	0.787080i								−0.476759	−	0.879034i	\(-0.658189\pi\)
							−0.995763	+	0.0919541i	\(0.970689\pi\)
\(7\)	−0.370976	−	0.0737916i	−0.140216	−	0.0278906i	0.124483	−	0.992222i	\(-0.460273\pi\)
							−0.264699	+	0.964331i	\(0.585273\pi\)
\(11\)	−4.94511	+	4.05835i	−1.49101	+	1.22364i	−0.577777	+	0.816195i	\(0.696080\pi\)
							−0.913230	+	0.407443i	\(0.866420\pi\)
\(13\)	2.85266	−	1.52478i	0.791186	−	0.422898i	−0.0257198	−	0.999669i	\(-0.508188\pi\)
							0.816905	+	0.576772i	\(0.195688\pi\)
\(17\)	1.84052	−	0.762368i	0.446391	−	0.184901i	−0.148153	−	0.988965i	\(-0.547333\pi\)
							0.594544	+	0.804063i	\(0.297333\pi\)
\(19\)	−1.60334	+	5.28549i	−0.367831	+	1.21258i	0.556456	+	0.830877i	\(0.312161\pi\)
							−0.924287	+	0.381698i	\(0.875339\pi\)
\(23\)	6.74914	+	4.50963i	1.40729	+	0.940323i	0.999630	+	0.0271906i	\(0.00865612\pi\)
							0.407663	+	0.913133i	\(0.366344\pi\)
\(29\)	−1.52102	−	1.24827i	−0.282447	−	0.231798i	0.482493	−	0.875900i	\(-0.339731\pi\)
							−0.764940	+	0.644102i	\(0.777231\pi\)
\(31\)	−0.401834	+	0.401834i	−0.0721716	+	0.0721716i	−0.742271	−	0.670100i	\(-0.766251\pi\)
							0.670100	+	0.742271i	\(0.266251\pi\)
\(37\)	1.65300	+	5.44922i	0.271752	+	0.895846i	0.981512	+	0.191402i	\(0.0613034\pi\)
							−0.709760	+	0.704444i	\(0.751197\pi\)
\(41\)	0.603568	+	0.403291i	0.0942614	+	0.0629835i	0.601808	−	0.798641i	\(-0.294447\pi\)
							−0.507546	+	0.861625i	\(0.669447\pi\)
\(43\)	0.837673	−	8.50504i	0.127744	−	1.29701i	−0.691003	−	0.722852i	\(-0.742831\pi\)
							0.818747	−	0.574155i	\(-0.194669\pi\)
\(47\)	0.573649	−	0.237613i	0.0836754	−	0.0346595i	−0.340453	−	0.940261i	\(-0.610580\pi\)
							0.424129	+	0.905602i	\(0.360580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.49	yes	992
3.2	odd	2	inner	384.2.w.a.11.14	✓	992
128.35	odd	32	inner	384.2.w.a.35.14	yes	992
384.35	even	32	inner	384.2.w.a.35.49	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.14	✓	992	3.2	odd	2	inner
384.2.w.a.11.49	yes	992	1.1	even	1	trivial
384.2.w.a.35.14	yes	992	128.35	odd	32	inner
384.2.w.a.35.49	yes	992	384.35	even	32	inner