Embedded newform 384.3.u.a.235.61

• Label: 384.3.u.a.235.61
• Level: $384$
• Weight: $3$
• Character: 384.235
• Analytic conductor: $10.463$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.u (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4632421514\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			235.61
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.98304 + 0.259877i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(3.86493 + 1.03069i) q^{4} +(2.50110 + 8.24503i) q^{5} +(-2.52692 + 2.36953i) q^{6} +(0.183376 - 0.921892i) q^{7} +(7.39647 + 3.04832i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.98304	+	0.259877i	0.991522	+	0.129938i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	2.50110	+	8.24503i	0.500221	+	1.64901i								0.735836	+	0.677160i	\(0.236790\pi\)
							−0.235615	+	0.971846i	\(0.575710\pi\)
\(7\)	0.183376	−	0.921892i	0.0261965	−	0.131699i	−0.965474	−	0.260500i	\(-0.916113\pi\)
							0.991671	+	0.128801i	\(0.0411127\pi\)
\(11\)	−8.93904	−	0.880418i	−0.812640	−	0.0800380i	−0.316845	−	0.948477i	\(-0.602623\pi\)
							−0.495795	+	0.868439i	\(0.665123\pi\)
\(13\)	18.8588	+	5.72075i	1.45068	+	0.440058i	0.914686	−	0.404166i	\(-0.132438\pi\)
							0.535991	+	0.844224i	\(0.319938\pi\)
\(17\)	−9.27660	−	22.3957i	−0.545682	−	1.31739i	−0.920662	−	0.390361i	\(-0.872350\pi\)
							0.374980	−	0.927033i	\(-0.377650\pi\)
\(19\)	−28.6074	+	15.2910i	−1.50565	+	0.804789i	−0.998375	−	0.0569918i	\(-0.981849\pi\)
							−0.507280	+	0.861781i	\(0.669349\pi\)
\(23\)	3.27595	+	2.18892i	0.142432	+	0.0951703i	0.624745	−	0.780829i	\(-0.285203\pi\)
							−0.482312	+	0.875999i	\(0.660203\pi\)
\(29\)	47.3146	−	4.66008i	1.63154	−	0.160693i	0.759628	−	0.650357i	\(-0.225381\pi\)
							0.871911	+	0.489665i	\(0.162881\pi\)
\(31\)	2.48190	−	2.48190i	0.0800612	−	0.0800612i	−0.665942	−	0.746003i	\(-0.731970\pi\)
							0.746003	+	0.665942i	\(0.231970\pi\)
\(37\)	−34.8095	−	18.6061i	−0.940798	−	0.502867i	−0.0716561	−	0.997429i	\(-0.522828\pi\)
							−0.869142	+	0.494562i	\(0.835328\pi\)
\(41\)	−13.4563	−	8.99122i	−0.328203	−	0.219298i	0.380541	−	0.924764i	\(-0.375738\pi\)
							−0.708744	+	0.705466i	\(0.750738\pi\)
\(43\)	40.2310	+	49.0216i	0.935604	+	1.14004i	0.989936	+	0.141514i	\(0.0451970\pi\)
							−0.0543321	+	0.998523i	\(0.517303\pi\)
\(47\)	−17.5184	−	42.2932i	−0.372732	−	0.899856i	−0.993285	−	0.115691i	\(-0.963092\pi\)
							0.620553	−	0.784165i	\(-0.286908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.61	yes	1024
128.67	odd	32	inner	384.3.u.a.67.61	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.61	✓	1024	128.67	odd	32	inner
384.3.u.a.235.61	yes	1024	1.1	even	1	trivial