Embedded newform 187.2.j.a.16.3

• Label: 187.2.j.a.16.3
• Level: $187$
• Weight: $2$
• Character: 187.16
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			16.3
Character	\(\chi\)	\(=\)	187.16
Dual form			187.2.j.a.152.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.609729 - 1.87655i) q^{2} +(-1.09362 + 1.50523i) q^{3} +(-1.53164 + 1.11281i) q^{4} +(-2.63951 - 0.857628i) q^{5} +(3.49146 + 1.13444i) q^{6} +(1.01174 + 1.39254i) q^{7} +(-0.170457 - 0.123844i) q^{8} +(-0.142682 - 0.439131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.609729	−	1.87655i	−0.431143	−	1.32692i	−0.896987	−	0.442056i	\(-0.854249\pi\)
							0.465844	−	0.884867i	\(-0.345751\pi\)
\(3\)	−1.09362	+	1.50523i	−0.631400	+	0.869048i	−0.998120	−	0.0612836i	\(-0.980481\pi\)
							0.366720	+	0.930331i	\(0.380481\pi\)
\(5\)	−2.63951	−	0.857628i	−1.18042	−	0.383543i	−0.347900	−	0.937532i	\(-0.613105\pi\)
							−0.832524	+	0.553989i	\(0.813105\pi\)
\(7\)	1.01174	+	1.39254i	0.382401	+	0.526330i	0.956219	−	0.292653i	\(-0.0945382\pi\)
							−0.573818	+	0.818983i	\(0.694538\pi\)
\(11\)	−0.792359	+	3.22058i	−0.238905	+	0.971043i
							\(13\)	1.73570	+	5.34194i	0.481397	+	1.48159i	0.837133	+	0.547000i	\(0.184230\pi\)
−0.355736	+	0.934586i	\(0.615770\pi\)
\(17\)	−0.469784	+	4.09625i	−0.113939	+	0.993488i
							\(19\)	−5.72692	−	4.16085i	−1.31385	−	0.954565i	−0.999987	−	0.00509576i	\(-0.998378\pi\)
−0.313859	−	0.949469i	\(-0.601622\pi\)
\(23\)		−	4.09281i		−	0.853411i	−0.904391	−	0.426705i	\(-0.859674\pi\)
							0.904391	−	0.426705i	\(-0.140326\pi\)
\(29\)	−3.82688	−	5.26725i	−0.710634	−	0.978104i	−0.999783	−	0.0208199i	\(-0.993372\pi\)
							0.289149	−	0.957284i	\(-0.406628\pi\)
\(31\)	0.631400	−	0.205154i	0.113403	−	0.0368468i	−0.251766	−	0.967788i	\(-0.581011\pi\)
							0.365169	+	0.930941i	\(0.381011\pi\)
\(37\)	5.21012	+	7.17111i	0.856538	+	1.17892i	0.982384	+	0.186874i	\(0.0598356\pi\)
							−0.125846	+	0.992050i	\(0.540164\pi\)
\(41\)	−4.35281	+	5.99113i	−0.679795	+	0.935658i	−0.999931	−	0.0117139i	\(-0.996271\pi\)
							0.320136	+	0.947371i	\(0.396271\pi\)
\(43\)	1.63319			0.249060			0.124530	−	0.992216i	\(-0.460258\pi\)
							0.124530	+	0.992216i	\(0.460258\pi\)
\(47\)	1.44404	+	1.04916i	0.210635	+	0.153036i	0.688101	−	0.725615i	\(-0.258445\pi\)
							−0.477466	+	0.878650i	\(0.658445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.16.3	✓	64
11.9	even	5	inner	187.2.j.a.152.4	yes	64
17.16	even	2	inner	187.2.j.a.16.4	yes	64
187.152	even	10	inner	187.2.j.a.152.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.16.3	✓	64	1.1	even	1	trivial
187.2.j.a.16.4	yes	64	17.16	even	2	inner
187.2.j.a.152.3	yes	64	187.152	even	10	inner
187.2.j.a.152.4	yes	64	11.9	even	5	inner