Embedded newform 165.2.r.a.74.19

• Label: 165.2.r.a.74.19
• 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			74.19
Character	\(\chi\)	\(=\)	165.74
Dual form			165.2.r.a.29.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.55596 + 2.14159i) q^{2} +(-1.61061 - 0.637133i) q^{3} +(-1.54738 + 4.76233i) q^{4} +(-1.59907 + 1.56301i) q^{5} +(-1.14156 - 4.44062i) q^{6} +(0.391826 - 1.20592i) q^{7} +(-7.57144 + 2.46011i) q^{8} +(2.18812 + 2.05234i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{4} - 10 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55596	+	2.14159i	1.10023	+	1.51433i	0.835082	+	0.550125i	\(0.185420\pi\)
							0.265146	+	0.964208i	\(0.414580\pi\)
\(3\)	−1.61061	−	0.637133i	−0.929886	−	0.367849i
							\(5\)	−1.59907	+	1.56301i	−0.715124	+	0.698997i
\(7\)	0.391826	−	1.20592i	0.148096	−	0.455793i								−0.849300	−	0.527911i	\(-0.822976\pi\)
							0.997396	+	0.0721173i	\(0.0229756\pi\)
\(11\)	2.72440	+	1.89146i	0.821439	+	0.570297i
							\(13\)	2.33830	−	1.69888i	0.648528	−	0.471183i	−0.214241	−	0.976781i	\(-0.568728\pi\)
0.862770	+	0.505597i	\(0.168728\pi\)
\(17\)	−2.46931	+	3.39871i	−0.598896	+	0.824309i	−0.995607	−	0.0936345i	\(-0.970151\pi\)
							0.396711	+	0.917944i	\(0.370151\pi\)
\(19\)	2.26276	−	0.735215i	0.519113	−	0.168670i	−0.0377299	−	0.999288i	\(-0.512013\pi\)
							0.556843	+	0.830618i	\(0.312013\pi\)
\(23\)	5.33159			1.11171			0.555857	−	0.831278i	\(-0.312390\pi\)
							0.555857	+	0.831278i	\(0.312390\pi\)
\(29\)	−0.920428	+	2.83279i	−0.170919	+	0.526035i	−0.999424	−	0.0339473i	\(-0.989192\pi\)
							0.828504	+	0.559982i	\(0.189192\pi\)
\(31\)	3.30827	−	2.40360i	0.594184	−	0.431700i	−0.249626	−	0.968342i	\(-0.580308\pi\)
							0.843810	+	0.536643i	\(0.180308\pi\)
\(37\)	−4.41557	−	1.43471i	−0.725915	−	0.235864i	−0.0773292	−	0.997006i	\(-0.524639\pi\)
							−0.648586	+	0.761142i	\(0.724639\pi\)
\(41\)	−2.88399	−	8.87601i	−0.450404	−	1.38620i	−0.876447	−	0.481498i	\(-0.840093\pi\)
							0.426043	−	0.904703i	\(-0.359907\pi\)
\(43\)	0.388853			0.0592995			0.0296498	−	0.999560i	\(-0.490561\pi\)
							0.0296498	+	0.999560i	\(0.490561\pi\)
\(47\)	−1.53724	−	4.73115i	−0.224230	−	0.690109i	−0.998369	−	0.0570931i	\(-0.981817\pi\)
							0.774139	−	0.633016i	\(-0.218183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.2.r.a.74.19	yes	80
3.2	odd	2	inner	165.2.r.a.74.1	yes	80
5.2	odd	4		825.2.bi.h.701.2		80
5.3	odd	4		825.2.bi.h.701.19		80
5.4	even	2	inner	165.2.r.a.74.2	yes	80
11.7	odd	10	inner	165.2.r.a.29.20	yes	80
15.2	even	4		825.2.bi.h.701.20		80
15.8	even	4		825.2.bi.h.701.1		80
15.14	odd	2	inner	165.2.r.a.74.20	yes	80
33.29	even	10	inner	165.2.r.a.29.2	yes	80
55.7	even	20		825.2.bi.h.326.20		80
55.18	even	20		825.2.bi.h.326.1		80
55.29	odd	10	inner	165.2.r.a.29.1	✓	80
165.29	even	10	inner	165.2.r.a.29.19	yes	80
165.62	odd	20		825.2.bi.h.326.2		80
165.128	odd	20		825.2.bi.h.326.19		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.r.a.29.1	✓	80	55.29	odd	10	inner
165.2.r.a.29.2	yes	80	33.29	even	10	inner
165.2.r.a.29.19	yes	80	165.29	even	10	inner
165.2.r.a.29.20	yes	80	11.7	odd	10	inner
165.2.r.a.74.1	yes	80	3.2	odd	2	inner
165.2.r.a.74.2	yes	80	5.4	even	2	inner
165.2.r.a.74.19	yes	80	1.1	even	1	trivial
165.2.r.a.74.20	yes	80	15.14	odd	2	inner
825.2.bi.h.326.1		80	55.18	even	20
825.2.bi.h.326.2		80	165.62	odd	20
825.2.bi.h.326.19		80	165.128	odd	20
825.2.bi.h.326.20		80	55.7	even	20
825.2.bi.h.701.1		80	15.8	even	4
825.2.bi.h.701.2		80	5.2	odd	4
825.2.bi.h.701.19		80	5.3	odd	4
825.2.bi.h.701.20		80	15.2	even	4