Embedded newform 384.3.u.a.235.10

• Label: 384.3.u.a.235.10
• 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			235.10
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78638 - 0.899353i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(2.38233 + 3.21318i) q^{4} +(-2.79381 - 9.20995i) q^{5} +(3.16702 - 1.40357i) q^{6} +(1.82226 - 9.16113i) q^{7} +(-1.36596 - 7.88252i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78638	−	0.899353i	−0.893191	−	0.449677i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	−2.79381	−	9.20995i	−0.558761	−	1.84199i								−0.539195	−	0.842181i	\(-0.681271\pi\)
							−0.0195663	−	0.999809i	\(-0.506229\pi\)
\(7\)	1.82226	−	9.16113i	0.260323	−	1.30873i	−0.600416	−	0.799688i	\(-0.704998\pi\)
							0.860739	−	0.509046i	\(-0.170002\pi\)
\(11\)	−14.2517	−	1.40367i	−1.29561	−	0.127606i	−0.573397	−	0.819277i	\(-0.694375\pi\)
							−0.722213	+	0.691671i	\(0.756875\pi\)
\(13\)	6.38443	+	1.93670i	0.491110	+	0.148977i	0.526120	−	0.850410i	\(-0.323646\pi\)
							−0.0350098	+	0.999387i	\(0.511146\pi\)
\(17\)	−2.61944	−	6.32388i	−0.154084	−	0.371993i	0.827921	−	0.560845i	\(-0.189523\pi\)
							−0.982006	+	0.188852i	\(0.939523\pi\)
\(19\)	27.7058	−	14.8091i	1.45820	−	0.779424i	0.463884	−	0.885896i	\(-0.346455\pi\)
							0.994316	+	0.106472i	\(0.0339555\pi\)
\(23\)	14.9089	+	9.96178i	0.648211	+	0.433121i	0.835730	−	0.549140i	\(-0.185045\pi\)
							−0.187519	+	0.982261i	\(0.560045\pi\)
\(29\)	−42.7768	+	4.21314i	−1.47506	+	0.145281i	−0.803339	−	0.595523i	\(-0.796945\pi\)
							−0.671722	+	0.740803i	\(0.734445\pi\)
\(31\)	−2.78591	+	2.78591i	−0.0898680	+	0.0898680i	−0.750612	−	0.660744i	\(-0.770241\pi\)
							0.660744	+	0.750612i	\(0.270241\pi\)
\(37\)	−17.1186	−	9.15010i	−0.462666	−	0.247300i	0.223558	−	0.974691i	\(-0.428233\pi\)
							−0.686224	+	0.727391i	\(0.740733\pi\)
\(41\)	0.790086	+	0.527918i	0.0192704	+	0.0128761i	0.565168	−	0.824976i	\(-0.308811\pi\)
							−0.545898	+	0.837852i	\(0.683811\pi\)
\(43\)	−6.02667	−	7.34352i	−0.140155	−	0.170780i	0.698202	−	0.715901i	\(-0.253984\pi\)
							−0.838357	+	0.545121i	\(0.816484\pi\)
\(47\)	26.1200	+	63.0593i	0.555745	+	1.34169i	0.913107	+	0.407721i	\(0.133676\pi\)
							−0.357362	+	0.933966i	\(0.616324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.10	yes	1024
128.67	odd	32	inner	384.3.u.a.67.10	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.10	✓	1024	128.67	odd	32	inner
384.3.u.a.235.10	yes	1024	1.1	even	1	trivial