Embedded newform 671.2.dc.a.6.4

• Label: 671.2.dc.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.56445 - 0.134397i) q^{2} +(-1.15775 + 0.376175i) q^{3} +(4.56929 + 0.480251i) q^{4} +(-1.75059 - 1.01070i) q^{5} +(3.01954 - 0.809083i) q^{6} +(3.73846 + 0.195925i) q^{7} +(-6.58045 - 1.04224i) q^{8} +(-1.22818 + 0.892324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.56445	−	0.134397i	−1.81334	−	0.0950330i	−0.884732	−	0.466100i	\(-0.845659\pi\)
							−0.928606	+	0.371067i	\(0.878992\pi\)
\(3\)	−1.15775	+	0.376175i	−0.668426	+	0.217185i	−0.623521	−	0.781807i	\(-0.714298\pi\)
							−0.0449048	+	0.998991i	\(0.514298\pi\)
\(5\)	−1.75059	−	1.01070i	−0.782888	−	0.452000i	0.0545650	−	0.998510i	\(-0.482623\pi\)
							−0.837453	+	0.546510i	\(0.815956\pi\)
\(7\)	3.73846	+	0.195925i	1.41301	+	0.0740525i	0.743491	−	0.668746i	\(-0.233168\pi\)
							0.669515	+	0.742798i	\(0.266502\pi\)
\(11\)	2.24645	+	2.43998i	0.677329	+	0.735680i
							\(13\)	0.319569	−	1.50345i	0.0886324	−	0.416983i	−0.911353	−	0.411625i	\(-0.864961\pi\)
0.999986	−	0.00535763i	\(-0.00170539\pi\)
\(17\)	−0.197265	+	0.736201i	−0.0478437	+	0.178555i	−0.985713	−	0.168434i	\(-0.946129\pi\)
							0.937869	+	0.346989i	\(0.112796\pi\)
\(19\)	−0.966840	+	1.07378i	−0.221808	+	0.246343i	−0.843770	−	0.536704i	\(-0.819669\pi\)
							0.621962	+	0.783047i	\(0.286336\pi\)
\(23\)	−0.0194625	+	0.0381973i	−0.00405822	+	0.00796470i	−0.893027	−	0.450004i	\(-0.851423\pi\)
							0.888969	+	0.457968i	\(0.151423\pi\)
\(29\)	−1.91698	−	4.99391i	−0.355974	−	0.927345i	−0.988284	−	0.152629i	\(-0.951226\pi\)
							0.632309	−	0.774716i	\(-0.282107\pi\)
\(31\)	4.30692	+	5.31860i	0.773546	+	0.955249i	0.999823	−	0.0188137i	\(-0.00598893\pi\)
							−0.226277	+	0.974063i	\(0.572656\pi\)
\(37\)	−2.61938	−	5.14082i	−0.430623	−	0.845145i	−0.999738	−	0.0228872i	\(-0.992714\pi\)
							0.569115	−	0.822258i	\(-0.307286\pi\)
\(41\)	−5.28648			−0.825610			−0.412805	−	0.910820i	\(-0.635451\pi\)
							−0.412805	+	0.910820i	\(0.635451\pi\)
\(43\)	−0.457760	+	8.73459i	−0.0698078	+	1.33201i	0.709204	+	0.705003i	\(0.249054\pi\)
							−0.779012	+	0.627009i	\(0.784279\pi\)
\(47\)	−0.0603885	+	0.574558i	−0.00880857	+	0.0838079i	−0.998043	−	0.0625317i	\(-0.980083\pi\)
							0.989234	+	0.146340i	\(0.0467492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.4	yes	960
11.2	odd	10		671.2.cu.a.189.4	✓	960
61.51	odd	60		671.2.cu.a.600.4	yes	960
671.112	even	60	inner	671.2.dc.a.112.4	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.4	✓	960	11.2	odd	10
671.2.cu.a.600.4	yes	960	61.51	odd	60
671.2.dc.a.6.4	yes	960	1.1	even	1	trivial
671.2.dc.a.112.4	yes	960	671.112	even	60	inner