Embedded newform 165.2.r.a.74.11

• Label: 165.2.r.a.74.11
• Level: $165$
• Weight: $2$
• Character: 165.74
• Analytic conductor: $1.318$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.31753163335\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			74.11
Character	\(\chi\)	\(=\)	165.74
Dual form			165.2.r.a.29.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.303223 + 0.417350i) q^{2} +(0.337382 + 1.69887i) q^{3} +(0.535797 - 1.64901i) q^{4} +(1.78836 - 1.34229i) q^{5} +(-0.606723 + 0.655944i) q^{6} +(0.781546 - 2.40535i) q^{7} +(1.83193 - 0.595230i) q^{8} +(-2.77235 + 1.14634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{4} - 10 q^{6} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(67\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.303223	+	0.417350i	0.214411	+	0.295111i	0.902652	−	0.430370i	\(-0.141617\pi\)
							−0.688242	+	0.725482i	\(0.741617\pi\)
\(3\)	0.337382	+	1.69887i	0.194788	+	0.980845i
							\(5\)	1.78836	−	1.34229i	0.799781	−	0.600292i
\(7\)	0.781546	−	2.40535i	0.295396	−	0.909137i								−0.687692	−	0.726003i	\(-0.741376\pi\)
							0.983088	−	0.183134i	\(-0.0586242\pi\)
\(11\)	−0.813609	+	3.21528i	−0.245312	+	0.969444i
							\(13\)	−4.13780	+	3.00629i	−1.14762	+	0.833794i	−0.988163	−	0.153411i	\(-0.950974\pi\)
−0.159457	+	0.987205i	\(0.550974\pi\)
\(17\)	−2.17976	+	3.00018i	−0.528668	+	0.727650i	−0.986927	−	0.161169i	\(-0.948473\pi\)
							0.458258	+	0.888819i	\(0.348473\pi\)
\(19\)	1.97662	−	0.642242i	0.453467	−	0.147340i	−0.0733735	−	0.997305i	\(-0.523377\pi\)
							0.526840	+	0.849964i	\(0.323377\pi\)
\(23\)	−2.64175			−0.550844			−0.275422	−	0.961323i	\(-0.588818\pi\)
							−0.275422	+	0.961323i	\(0.588818\pi\)
\(29\)	−1.79775	+	5.53289i	−0.333833	+	1.02743i	0.633461	+	0.773775i	\(0.281634\pi\)
							−0.967294	+	0.253658i	\(0.918366\pi\)
\(31\)	4.64730	−	3.37646i	0.834680	−	0.606431i	−0.0861994	−	0.996278i	\(-0.527472\pi\)
							0.920879	+	0.389847i	\(0.127472\pi\)
\(37\)	8.16423	+	2.65272i	1.34219	+	0.436104i	0.890059	−	0.455846i	\(-0.150663\pi\)
							0.452133	+	0.891951i	\(0.350663\pi\)
\(41\)	−0.856736	−	2.63676i	−0.133800	−	0.411793i	0.861602	−	0.507585i	\(-0.169462\pi\)
							−0.995401	+	0.0957919i	\(0.969462\pi\)
\(43\)	−8.05145			−1.22783			−0.613917	−	0.789370i	\(-0.710407\pi\)
							−0.613917	+	0.789370i	\(0.710407\pi\)
\(47\)	−1.10039	−	3.38664i	−0.160508	−	0.493993i	0.838169	−	0.545410i	\(-0.183626\pi\)
							−0.998677	+	0.0514175i	\(0.983626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.2.r.a.74.11	yes	80
3.2	odd	2	inner	165.2.r.a.74.9	yes	80
5.2	odd	4		825.2.bi.h.701.10		80
5.3	odd	4		825.2.bi.h.701.11		80
5.4	even	2	inner	165.2.r.a.74.10	yes	80
11.7	odd	10	inner	165.2.r.a.29.12	yes	80
15.2	even	4		825.2.bi.h.701.12		80
15.8	even	4		825.2.bi.h.701.9		80
15.14	odd	2	inner	165.2.r.a.74.12	yes	80
33.29	even	10	inner	165.2.r.a.29.10	yes	80
55.7	even	20		825.2.bi.h.326.12		80
55.18	even	20		825.2.bi.h.326.9		80
55.29	odd	10	inner	165.2.r.a.29.9	✓	80
165.29	even	10	inner	165.2.r.a.29.11	yes	80
165.62	odd	20		825.2.bi.h.326.10		80
165.128	odd	20		825.2.bi.h.326.11		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.r.a.29.9	✓	80	55.29	odd	10	inner
165.2.r.a.29.10	yes	80	33.29	even	10	inner
165.2.r.a.29.11	yes	80	165.29	even	10	inner
165.2.r.a.29.12	yes	80	11.7	odd	10	inner
165.2.r.a.74.9	yes	80	3.2	odd	2	inner
165.2.r.a.74.10	yes	80	5.4	even	2	inner
165.2.r.a.74.11	yes	80	1.1	even	1	trivial
165.2.r.a.74.12	yes	80	15.14	odd	2	inner
825.2.bi.h.326.9		80	55.18	even	20
825.2.bi.h.326.10		80	165.62	odd	20
825.2.bi.h.326.11		80	165.128	odd	20
825.2.bi.h.326.12		80	55.7	even	20
825.2.bi.h.701.9		80	15.8	even	4
825.2.bi.h.701.10		80	5.2	odd	4
825.2.bi.h.701.11		80	5.3	odd	4
825.2.bi.h.701.12		80	15.2	even	4