Embedded newform 671.2.dc.a.6.2

• Label: 671.2.dc.a.6.2
• Level: $671$
• Weight: $2$
• Character: 671.6
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $960$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.dc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(60\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			6.2
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74850 - 0.144043i) q^{2} +(-0.775665 + 0.252029i) q^{3} +(5.54446 + 0.582747i) q^{4} +(-0.282415 - 0.163052i) q^{5} +(2.16822 - 0.580972i) q^{6} +(-4.98173 - 0.261081i) q^{7} +(-9.71824 - 1.53922i) q^{8} +(-1.88891 + 1.37238i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{60}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74850	−	0.144043i	−1.94348	−	0.101854i	−0.959679	−	0.281099i	\(-0.909301\pi\)
							−0.983804	+	0.179245i	\(0.942635\pi\)
\(3\)	−0.775665	+	0.252029i	−0.447830	+	0.145509i	−0.524245	−	0.851567i	\(-0.675653\pi\)
							0.0764152	+	0.997076i	\(0.475653\pi\)
\(5\)	−0.282415	−	0.163052i	−0.126300	−	0.0729191i	0.435519	−	0.900179i	\(-0.356565\pi\)
							−0.561819	+	0.827260i	\(0.689898\pi\)
\(7\)	−4.98173	−	0.261081i	−1.88292	−	0.0986795i	−0.924004	−	0.382382i	\(-0.875104\pi\)
							−0.958912	+	0.283702i	\(0.908437\pi\)
\(11\)	−3.09242	+	1.19874i	−0.932398	+	0.361433i
							\(13\)	1.01735	−	4.78625i	0.282162	−	1.32747i	−0.577411	−	0.816454i	\(-0.695937\pi\)
0.859573	−	0.511014i	\(-0.170730\pi\)
\(17\)	−0.650809	+	2.42885i	−0.157844	+	0.589083i	0.841001	+	0.541034i	\(0.181967\pi\)
							−0.998845	+	0.0480490i	\(0.984700\pi\)
\(19\)	−0.255481	+	0.283741i	−0.0586114	+	0.0650946i	−0.771737	−	0.635941i	\(-0.780612\pi\)
							0.713126	+	0.701036i	\(0.247279\pi\)
\(23\)	0.417095	−	0.818595i	0.0869703	−	0.170689i	−0.843425	−	0.537248i	\(-0.819464\pi\)
							0.930395	+	0.366559i	\(0.119464\pi\)
\(29\)	2.31723	+	6.03658i	0.430298	+	1.12097i	0.961763	+	0.273885i	\(0.0883086\pi\)
							−0.531464	+	0.847081i	\(0.678358\pi\)
\(31\)	0.750112	+	0.926312i	0.134724	+	0.166371i	0.840043	−	0.542520i	\(-0.182530\pi\)
							−0.705319	+	0.708890i	\(0.749196\pi\)
\(37\)	3.78507	+	7.42861i	0.622261	+	1.22126i	0.959996	+	0.280014i	\(0.0903393\pi\)
							−0.337735	+	0.941241i	\(0.609661\pi\)
\(41\)	−6.86254			−1.07175			−0.535874	−	0.844298i	\(-0.680018\pi\)
							−0.535874	+	0.844298i	\(0.680018\pi\)
\(43\)	0.558107	−	10.6493i	0.0851105	−	1.62401i	−0.538747	−	0.842467i	\(-0.681102\pi\)
							0.623858	−	0.781538i	\(-0.285564\pi\)
\(47\)	0.138524	−	1.31797i	0.0202058	−	0.192245i	−0.979763	−	0.200162i	\(-0.935853\pi\)
							0.999969	+	0.00791644i	\(0.00251991\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.2	yes	960
11.2	odd	10		671.2.cu.a.189.2	✓	960
61.51	odd	60		671.2.cu.a.600.2	yes	960
671.112	even	60	inner	671.2.dc.a.112.2	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.2	✓	960	11.2	odd	10
671.2.cu.a.600.2	yes	960	61.51	odd	60
671.2.dc.a.6.2	yes	960	1.1	even	1	trivial
671.2.dc.a.112.2	yes	960	671.112	even	60	inner