Embedded newform 381.2.x.b.245.7

• Label: 381.2.x.b.245.7
• Level: $381$
• Weight: $2$
• Character: 381.245
• Analytic conductor: $3.042$
• Analytic rank: $0$
• Dimension: $1440$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	381.x (of order \(126\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.04230031701\)
Analytic rank:	\(0\)
Dimension:	\(1440\)
Relative dimension:	\(40\) over \(\Q(\zeta_{126})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{126}]$

Embedding invariants

Embedding label			245.7
Character	\(\chi\)	\(=\)	381.245
Dual form			381.2.x.b.14.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.921731 + 1.91399i) q^{2} +(-1.02218 + 1.39826i) q^{3} +(-1.56681 - 1.96471i) q^{4} +(0.361993 + 0.922343i) q^{5} +(-1.73409 - 3.24528i) q^{6} +(2.39642 + 3.92156i) q^{7} +(1.06240 - 0.242486i) q^{8} +(-0.910283 - 2.85856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 36 q^{3} + 192 q^{4} - 57 q^{6} - 78 q^{7} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/381\mathbb{Z}\right)^\times\).

\(n\)	\(128\)	\(130\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{65}{126}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.921731	+	1.91399i	−0.651762	+	1.35340i	0.268949	+	0.963154i	\(0.413324\pi\)
							−0.920712	+	0.390244i	\(0.872391\pi\)
\(3\)	−1.02218	+	1.39826i	−0.590158	+	0.807288i
							\(5\)	0.361993	+	0.922343i	0.161888	+	0.412485i	0.988683	−	0.150019i	\(-0.0479335\pi\)
−0.826795	+	0.562504i	\(0.809838\pi\)
\(7\)	2.39642	+	3.92156i	0.905760	+	1.48221i	0.875808	+	0.482660i	\(0.160329\pi\)
							0.0299527	+	0.999551i	\(0.490464\pi\)
\(11\)	−1.28161	+	5.03319i	−0.386419	+	1.51756i	0.409584	+	0.912272i	\(0.365674\pi\)
							−0.796004	+	0.605292i	\(0.793056\pi\)
\(13\)	0.0202101	+	0.810401i	0.00560528	+	0.224765i	0.996155	+	0.0876097i	\(0.0279228\pi\)
							−0.990550	+	0.137155i	\(0.956204\pi\)
\(17\)	−2.39567	−	0.804903i	−0.581036	−	0.195218i	0.0126424	−	0.999920i	\(-0.495976\pi\)
							−0.593678	+	0.804702i	\(0.702325\pi\)
\(19\)	−0.894427	−	1.54919i	−0.205196	−	0.355409i	0.744999	−	0.667065i	\(-0.232450\pi\)
							−0.950195	+	0.311656i	\(0.899116\pi\)
\(23\)	4.33357	−	1.10346i	0.903612	−	0.230088i	0.233508	−	0.972355i	\(-0.424980\pi\)
							0.670104	+	0.742267i	\(0.266249\pi\)
\(29\)	6.00618	−	4.54950i	1.11532	−	0.844822i	0.126210	−	0.992004i	\(-0.459719\pi\)
							0.989109	+	0.147182i	\(0.0470203\pi\)
\(31\)	−5.55382	+	6.29325i	−0.997496	+	1.13030i	−0.00602365	+	0.999982i	\(0.501917\pi\)
							−0.991472	+	0.130318i	\(0.958400\pi\)
\(37\)	0.497179	−	2.81964i	0.0817357	−	0.463546i	−0.916278	−	0.400544i	\(-0.868821\pi\)
							0.998013	−	0.0630026i	\(-0.0200677\pi\)
\(41\)	−3.79143	+	3.34596i	−0.592123	+	0.522551i	−0.903978	−	0.427578i	\(-0.859367\pi\)
							0.311856	+	0.950129i	\(0.399049\pi\)
\(43\)	1.84762	−	4.38307i	0.281759	−	0.668412i	−0.717855	−	0.696193i	\(-0.754876\pi\)
							0.999614	+	0.0277809i	\(0.00884408\pi\)
\(47\)	−0.280575	−	1.86149i	−0.0409261	−	0.271527i	0.959004	−	0.283394i	\(-0.0914605\pi\)
							−0.999930	+	0.0118675i	\(0.996222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	381.2.x.b.245.7	yes	1440
3.2	odd	2	inner	381.2.x.b.245.34	yes	1440
127.14	odd	126	inner	381.2.x.b.14.34	yes	1440
381.14	even	126	inner	381.2.x.b.14.7	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.x.b.14.7	✓	1440	381.14	even	126	inner
381.2.x.b.14.34	yes	1440	127.14	odd	126	inner
381.2.x.b.245.7	yes	1440	1.1	even	1	trivial
381.2.x.b.245.34	yes	1440	3.2	odd	2	inner