Embedded newform 384.3.u.a.235.35

• Label: 384.3.u.a.235.35
• 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			235.35
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.350937 + 1.96897i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(-3.75369 + 1.38197i) q^{4} +(0.819281 + 2.70081i) q^{5} +(-3.02185 - 1.69364i) q^{6} +(1.27192 - 6.39436i) q^{7} +(-4.03836 - 6.90591i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.350937	+	1.96897i	0.175469	+	0.984485i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	0.819281	+	2.70081i	0.163856	+	0.540162i								0.999965	−	0.00840247i	\(-0.00267462\pi\)
							−0.836108	+	0.548564i	\(0.815175\pi\)
\(7\)	1.27192	−	6.39436i	0.181702	−	0.913480i	−0.777094	−	0.629385i	\(-0.783307\pi\)
							0.958796	−	0.284095i	\(-0.0916931\pi\)
\(11\)	−11.7255	−	1.15486i	−1.06595	−	0.104987i	−0.450202	−	0.892927i	\(-0.648648\pi\)
							−0.615753	+	0.787939i	\(0.711148\pi\)
\(13\)	−1.27687	−	0.387334i	−0.0982207	−	0.0297949i	0.240792	−	0.970577i	\(-0.422593\pi\)
							−0.339013	+	0.940782i	\(0.610093\pi\)
\(17\)	−9.37411	−	22.6311i	−0.551418	−	1.33124i	−0.916414	−	0.400232i	\(-0.868929\pi\)
							0.364995	−	0.931009i	\(-0.381071\pi\)
\(19\)	−1.67623	+	0.895963i	−0.0882226	+	0.0471560i	−0.514918	−	0.857239i	\(-0.672178\pi\)
							0.426695	+	0.904395i	\(0.359678\pi\)
\(23\)	−14.4456	−	9.65227i	−0.628071	−	0.419664i	0.200385	−	0.979717i	\(-0.435781\pi\)
							−0.828457	+	0.560053i	\(0.810781\pi\)
\(29\)	−13.6957	+	1.34891i	−0.472267	+	0.0465142i	−0.331351	−	0.943507i	\(-0.607505\pi\)
							−0.140915	+	0.990022i	\(0.545005\pi\)
\(31\)	−10.1717	+	10.1717i	−0.328119	+	0.328119i	−0.851871	−	0.523752i	\(-0.824532\pi\)
							0.523752	+	0.851871i	\(0.324532\pi\)
\(37\)	43.1849	+	23.0828i	1.16716	+	0.623859i	0.937012	−	0.349297i	\(-0.113580\pi\)
							0.230146	+	0.973156i	\(0.426080\pi\)
\(41\)	5.45435	+	3.64448i	0.133033	+	0.0888898i	0.620306	−	0.784360i	\(-0.287008\pi\)
							−0.487273	+	0.873250i	\(0.662008\pi\)
\(43\)	−9.48192	−	11.5538i	−0.220510	−	0.268692i	0.650992	−	0.759085i	\(-0.274353\pi\)
							−0.871501	+	0.490393i	\(0.836853\pi\)
\(47\)	−18.7295	−	45.2169i	−0.398499	−	0.962063i	−0.988022	−	0.154311i	\(-0.950684\pi\)
							0.589523	−	0.807752i	\(-0.299316\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.35	yes	1024
128.67	odd	32	inner	384.3.u.a.67.35	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.35	✓	1024	128.67	odd	32	inner
384.3.u.a.235.35	yes	1024	1.1	even	1	trivial