Embedded newform 384.2.w.a.35.7

• Label: 384.2.w.a.35.7
• 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			35.7
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30056 + 0.555477i) q^{2} +(-1.67960 - 0.423001i) q^{3} +(1.38289 - 1.44486i) q^{4} +(3.35351 - 1.79249i) q^{5} +(2.41939 - 0.382846i) q^{6} +(3.20524 - 0.637561i) q^{7} +(-0.995942 + 2.64728i) q^{8} +(2.64214 + 1.42095i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30056	+	0.555477i	−0.919632	+	0.392782i
							\(3\)	−1.67960	−	0.423001i	−0.969720	−	0.244220i
\(5\)	3.35351	−	1.79249i	1.49973	−	0.801624i								0.501745	−	0.865016i	\(-0.332692\pi\)
							0.997989	+	0.0633915i	\(0.0201917\pi\)
\(7\)	3.20524	−	0.637561i	1.21147	−	0.240975i	0.452295	−	0.891868i	\(-0.350605\pi\)
							0.759170	+	0.650893i	\(0.225605\pi\)
\(11\)	−2.82151	−	2.31556i	−0.850719	−	0.698167i	0.104353	−	0.994540i	\(-0.466723\pi\)
							−0.955072	+	0.296373i	\(0.904223\pi\)
\(13\)	−3.67615	−	1.96494i	−1.01958	−	0.544977i	−0.125128	−	0.992141i	\(-0.539934\pi\)
							−0.894452	+	0.447163i	\(0.852434\pi\)
\(17\)	4.87094	+	2.01761i	1.18138	+	0.489342i	0.884937	−	0.465710i	\(-0.154201\pi\)
							0.296438	+	0.955052i	\(0.404201\pi\)
\(19\)	1.12265	+	3.70087i	0.257553	+	0.849039i	0.986400	+	0.164362i	\(0.0525565\pi\)
							−0.728847	+	0.684677i	\(0.759943\pi\)
\(23\)	−0.487940	+	0.326031i	−0.101743	+	0.0679822i	−0.605401	−	0.795921i	\(-0.706987\pi\)
							0.503658	+	0.863903i	\(0.331987\pi\)
\(29\)	4.48972	−	3.68462i	0.833720	−	0.684217i	−0.117384	−	0.993087i	\(-0.537451\pi\)
							0.951104	+	0.308870i	\(0.0999508\pi\)
\(31\)	−4.98746	−	4.98746i	−0.895774	−	0.895774i	0.0992850	−	0.995059i	\(-0.468344\pi\)
							−0.995059	+	0.0992850i	\(0.968344\pi\)
\(37\)	−2.84407	+	9.37564i	−0.467562	+	1.54135i	0.333492	+	0.942753i	\(0.391773\pi\)
							−0.801054	+	0.598592i	\(0.795727\pi\)
\(41\)	6.15597	−	4.11328i	0.961400	−	0.642387i	0.0273868	−	0.999625i	\(-0.491281\pi\)
							0.934013	+	0.357238i	\(0.116281\pi\)
\(43\)	0.0174813	+	0.177490i	0.00266587	+	0.0270670i	0.996426	−	0.0844746i	\(-0.0269212\pi\)
							−0.993760	+	0.111542i	\(0.964421\pi\)
\(47\)	−0.894161	−	0.370374i	−0.130427	−	0.0540246i	0.316515	−	0.948587i	\(-0.397487\pi\)
							−0.446942	+	0.894563i	\(0.647487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.35.7	yes	992
3.2	odd	2	inner	384.2.w.a.35.56	yes	992
128.11	odd	32	inner	384.2.w.a.11.56	yes	992
384.11	even	32	inner	384.2.w.a.11.7	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.7	✓	992	384.11	even	32	inner
384.2.w.a.11.56	yes	992	128.11	odd	32	inner
384.2.w.a.35.7	yes	992	1.1	even	1	trivial
384.2.w.a.35.56	yes	992	3.2	odd	2	inner