Embedded newform 165.4.w.a.118.18

• Label: 165.4.w.a.118.18
• Level: $165$
• Weight: $4$
• Character: 165.118
• Analytic conductor: $9.735$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.w (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.73531515095\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			118.18
Character	\(\chi\)	\(=\)	165.118
Dual form			165.4.w.a.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.623925 - 0.0988201i) q^{2} +(-1.36197 + 2.67302i) q^{3} +(-7.22893 - 2.34882i) q^{4} +(9.11856 + 6.46930i) q^{5} +(1.11392 - 1.53317i) q^{6} +(19.1840 - 9.77474i) q^{7} +(8.78101 + 4.47415i) q^{8} +(-5.29007 - 7.28115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q + 32 q^{5} - 40 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(46\)	\(56\)	\(67\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.623925	−	0.0988201i	−0.220591	−	0.0349382i	0.0451606	−	0.998980i	\(-0.485620\pi\)
							−0.265751	+	0.964042i	\(0.585620\pi\)
\(3\)	−1.36197	+	2.67302i	−0.262112	+	0.514423i
							\(5\)	9.11856	+	6.46930i	0.815589	+	0.578632i
\(7\)	19.1840	−	9.77474i	1.03584	−	0.527786i								0.148504	−	0.988912i	\(-0.452554\pi\)
							0.887335	+	0.461125i	\(0.152554\pi\)
\(11\)	−36.4611	+	1.26076i	−0.999403	+	0.0345576i
							\(13\)	3.87521	−	24.4671i	0.0826761	−	0.521996i	−0.911242	−	0.411872i	\(-0.864875\pi\)
0.993918	−	0.110124i	\(-0.0351249\pi\)
\(17\)	15.1873	+	95.8888i	0.216674	+	1.36803i	0.820835	+	0.571165i	\(0.193508\pi\)
							−0.604161	+	0.796862i	\(0.706492\pi\)
\(19\)	29.3757	+	90.4091i	0.354697	+	1.09165i	0.956185	+	0.292764i	\(0.0945750\pi\)
							−0.601487	+	0.798882i	\(0.705425\pi\)
\(23\)	123.152	+	123.152i	1.11648	+	1.11648i	0.992254	+	0.124226i	\(0.0396449\pi\)
							0.124226	+	0.992254i	\(0.460355\pi\)
\(29\)	−31.8296	+	97.9616i	−0.203814	+	0.627276i	0.795946	+	0.605368i	\(0.206974\pi\)
							−0.999760	+	0.0219080i	\(0.993026\pi\)
\(31\)	−6.98693	+	5.07630i	−0.0404803	+	0.0294106i	−0.607841	−	0.794058i	\(-0.707964\pi\)
							0.567361	+	0.823469i	\(0.307964\pi\)
\(37\)	−112.056	−	219.923i	−0.497891	−	0.977166i	−0.994049	−	0.108934i	\(-0.965256\pi\)
							0.496158	−	0.868232i	\(-0.334744\pi\)
\(41\)	143.602	−	46.6592i	0.546997	−	0.177730i	−0.0224651	−	0.999748i	\(-0.507151\pi\)
							0.569462	+	0.822017i	\(0.307151\pi\)
\(43\)	−343.719	+	343.719i	−1.21899	+	1.21899i	−0.251008	+	0.967985i	\(0.580762\pi\)
							−0.967985	+	0.251008i	\(0.919238\pi\)
\(47\)	−128.308	−	65.3762i	−0.398205	−	0.202896i	0.243408	−	0.969924i	\(-0.421735\pi\)
							−0.641613	+	0.767028i	\(0.721735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.w.a.118.18	yes	288
5.2	odd	4	inner	165.4.w.a.52.19	yes	288
11.7	odd	10	inner	165.4.w.a.73.19	yes	288
55.7	even	20	inner	165.4.w.a.7.18	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.w.a.7.18	✓	288	55.7	even	20	inner
165.4.w.a.52.19	yes	288	5.2	odd	4	inner
165.4.w.a.73.19	yes	288	11.7	odd	10	inner
165.4.w.a.118.18	yes	288	1.1	even	1	trivial