Embedded newform 384.2.w.a.35.14

• Label: 384.2.w.a.35.14
• 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			35.14
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11401 + 0.871200i) q^{2} +(0.593110 - 1.62734i) q^{3} +(0.482021 - 1.94104i) q^{4} +(3.29266 - 1.75996i) q^{5} +(0.757006 + 2.32958i) q^{6} +(-0.370976 + 0.0737916i) q^{7} +(1.15406 + 2.58227i) q^{8} +(-2.29644 - 1.93038i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11401	+	0.871200i	−0.787722	+	0.616031i
							\(3\)	0.593110	−	1.62734i	0.342432	−	0.939543i
\(5\)	3.29266	−	1.75996i	1.47252	−	0.787080i								0.476759	−	0.879034i	\(-0.341811\pi\)
							0.995763	+	0.0919541i	\(0.0293113\pi\)
\(7\)	−0.370976	+	0.0737916i	−0.140216	+	0.0278906i	−0.264699	−	0.964331i	\(-0.585273\pi\)
							0.124483	+	0.992222i	\(0.460273\pi\)
\(11\)	4.94511	+	4.05835i	1.49101	+	1.22364i	0.913230	+	0.407443i	\(0.133580\pi\)
							0.577777	+	0.816195i	\(0.303920\pi\)
\(13\)	2.85266	+	1.52478i	0.791186	+	0.422898i	0.816905	−	0.576772i	\(-0.195688\pi\)
							−0.0257198	+	0.999669i	\(0.508188\pi\)
\(17\)	−1.84052	−	0.762368i	−0.446391	−	0.184901i	0.148153	−	0.988965i	\(-0.452667\pi\)
							−0.594544	+	0.804063i	\(0.702667\pi\)
\(19\)	−1.60334	−	5.28549i	−0.367831	−	1.21258i	−0.924287	−	0.381698i	\(-0.875339\pi\)
							0.556456	−	0.830877i	\(-0.312161\pi\)
\(23\)	−6.74914	+	4.50963i	−1.40729	+	0.940323i	−0.407663	+	0.913133i	\(0.633656\pi\)
							−0.999630	+	0.0271906i	\(0.991344\pi\)
\(29\)	1.52102	−	1.24827i	0.282447	−	0.231798i	−0.482493	−	0.875900i	\(-0.660269\pi\)
							0.764940	+	0.644102i	\(0.222769\pi\)
\(31\)	−0.401834	−	0.401834i	−0.0721716	−	0.0721716i	0.670100	−	0.742271i	\(-0.266251\pi\)
							−0.742271	+	0.670100i	\(0.766251\pi\)
\(37\)	1.65300	−	5.44922i	0.271752	−	0.895846i	−0.709760	−	0.704444i	\(-0.751197\pi\)
							0.981512	−	0.191402i	\(-0.0613034\pi\)
\(41\)	−0.603568	+	0.403291i	−0.0942614	+	0.0629835i	−0.601808	−	0.798641i	\(-0.705553\pi\)
							0.507546	+	0.861625i	\(0.330553\pi\)
\(43\)	0.837673	+	8.50504i	0.127744	+	1.29701i	0.818747	+	0.574155i	\(0.194669\pi\)
							−0.691003	+	0.722852i	\(0.742831\pi\)
\(47\)	−0.573649	−	0.237613i	−0.0836754	−	0.0346595i	0.340453	−	0.940261i	\(-0.389420\pi\)
							−0.424129	+	0.905602i	\(0.639420\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.14	✓	992	384.11	even	32	inner
384.2.w.a.11.49	yes	992	128.11	odd	32	inner
384.2.w.a.35.14	yes	992	1.1	even	1	trivial
384.2.w.a.35.49	yes	992	3.2	odd	2	inner