Embedded newform 165.2.r.a.74.8

• Label: 165.2.r.a.74.8
• 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			74.8
Character	\(\chi\)	\(=\)	165.74
Dual form			165.2.r.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.413334 - 0.568905i) q^{2} +(1.71421 - 0.247937i) q^{3} +(0.465226 - 1.43182i) q^{4} +(2.00721 + 0.985452i) q^{5} +(-0.849595 - 0.872744i) q^{6} +(-0.615368 + 1.89391i) q^{7} +(-2.34444 + 0.761754i) q^{8} +(2.87705 - 0.850033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{4} - 10 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.413334	−	0.568905i	−0.292271	−	0.402277i	0.637479	−	0.770468i	\(-0.279977\pi\)
							−0.929750	+	0.368191i	\(0.879977\pi\)
\(3\)	1.71421	−	0.247937i	0.989702	−	0.143146i
							\(5\)	2.00721	+	0.985452i	0.897651	+	0.440707i
\(7\)	−0.615368	+	1.89391i	−0.232587	+	0.715830i								0.764845	+	0.644214i	\(0.222816\pi\)
							−0.997432	+	0.0716158i	\(0.977184\pi\)
\(11\)	−3.26241	−	0.597209i	−0.983655	−	0.180065i
							\(13\)	−3.43403	+	2.49497i	−0.952429	+	0.691980i	−0.951380	−	0.308019i	\(-0.900334\pi\)
−0.00104920	+	0.999999i	\(0.500334\pi\)
\(17\)	2.51291	−	3.45873i	0.609471	−	0.838864i	−0.387063	−	0.922053i	\(-0.626510\pi\)
							0.996534	+	0.0831888i	\(0.0265105\pi\)
\(19\)	0.257486	−	0.0836623i	0.0590714	−	0.0191935i	−0.279332	−	0.960195i	\(-0.590113\pi\)
							0.338404	+	0.941001i	\(0.390113\pi\)
\(23\)	−4.30856			−0.898397			−0.449198	−	0.893432i	\(-0.648290\pi\)
							−0.449198	+	0.893432i	\(0.648290\pi\)
\(29\)	2.26468	−	6.96997i	0.420540	−	1.29429i	−0.486660	−	0.873591i	\(-0.661785\pi\)
							0.907200	−	0.420699i	\(-0.138215\pi\)
\(31\)	−5.24479	+	3.81056i	−0.941992	+	0.684397i	−0.948899	−	0.315579i	\(-0.897801\pi\)
							0.00690742	+	0.999976i	\(0.497801\pi\)
\(37\)	−5.66863	−	1.84185i	−0.931917	−	0.302798i	−0.196571	−	0.980490i	\(-0.562980\pi\)
							−0.735346	+	0.677692i	\(0.762980\pi\)
\(41\)	2.95108	+	9.08248i	0.460881	+	1.41844i	0.864090	+	0.503337i	\(0.167895\pi\)
							−0.403210	+	0.915108i	\(0.632105\pi\)
\(43\)	1.68949			0.257645			0.128822	−	0.991668i	\(-0.458880\pi\)
							0.128822	+	0.991668i	\(0.458880\pi\)
\(47\)	−0.0723398	−	0.222639i	−0.0105518	−	0.0324752i	0.945642	−	0.325210i	\(-0.105435\pi\)
							−0.956194	+	0.292734i	\(0.905435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.2.r.a.74.8	yes	80
3.2	odd	2	inner	165.2.r.a.74.14	yes	80
5.2	odd	4		825.2.bi.h.701.13		80
5.3	odd	4		825.2.bi.h.701.8		80
5.4	even	2	inner	165.2.r.a.74.13	yes	80
11.7	odd	10	inner	165.2.r.a.29.7	✓	80
15.2	even	4		825.2.bi.h.701.7		80
15.8	even	4		825.2.bi.h.701.14		80
15.14	odd	2	inner	165.2.r.a.74.7	yes	80
33.29	even	10	inner	165.2.r.a.29.13	yes	80
55.7	even	20		825.2.bi.h.326.7		80
55.18	even	20		825.2.bi.h.326.14		80
55.29	odd	10	inner	165.2.r.a.29.14	yes	80
165.29	even	10	inner	165.2.r.a.29.8	yes	80
165.62	odd	20		825.2.bi.h.326.13		80
165.128	odd	20		825.2.bi.h.326.8		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.r.a.29.7	✓	80	11.7	odd	10	inner
165.2.r.a.29.8	yes	80	165.29	even	10	inner
165.2.r.a.29.13	yes	80	33.29	even	10	inner
165.2.r.a.29.14	yes	80	55.29	odd	10	inner
165.2.r.a.74.7	yes	80	15.14	odd	2	inner
165.2.r.a.74.8	yes	80	1.1	even	1	trivial
165.2.r.a.74.13	yes	80	5.4	even	2	inner
165.2.r.a.74.14	yes	80	3.2	odd	2	inner
825.2.bi.h.326.7		80	55.7	even	20
825.2.bi.h.326.8		80	165.128	odd	20
825.2.bi.h.326.13		80	165.62	odd	20
825.2.bi.h.326.14		80	55.18	even	20
825.2.bi.h.701.7		80	15.2	even	4
825.2.bi.h.701.8		80	5.3	odd	4
825.2.bi.h.701.13		80	5.2	odd	4
825.2.bi.h.701.14		80	15.8	even	4