Embedded newform 187.2.r.a.15.2

• Label: 187.2.r.a.15.2
• Level: $187$
• Weight: $2$
• Character: 187.15
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.r (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.2
Character	\(\chi\)	\(=\)	187.15
Dual form			187.2.r.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.368659 - 2.32762i) q^{2} +(-0.913416 + 0.0718874i) q^{3} +(-3.37979 + 1.09816i) q^{4} +(1.97221 - 3.21836i) q^{5} +(0.504065 + 2.09958i) q^{6} +(0.151479 - 1.92472i) q^{7} +(1.66231 + 3.26246i) q^{8} +(-2.13390 + 0.337977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.368659	−	2.32762i	−0.260681	−	1.64588i	−0.676511	−	0.736433i	\(-0.736509\pi\)
							0.415830	−	0.909443i	\(-0.363491\pi\)
\(3\)	−0.913416	+	0.0718874i	−0.527361	+	0.0415042i	−0.339345	−	0.940662i	\(-0.610206\pi\)
							−0.188016	+	0.982166i	\(0.560206\pi\)
\(5\)	1.97221	−	3.21836i	0.882001	−	1.43930i	−0.0162767	−	0.999868i	\(-0.505181\pi\)
							0.898278	−	0.439428i	\(-0.144819\pi\)
\(7\)	0.151479	−	1.92472i	0.0572537	−	0.727477i	−0.899564	−	0.436789i	\(-0.856116\pi\)
							0.956818	−	0.290688i	\(-0.0938843\pi\)
\(11\)	2.65129	+	1.99265i	0.799395	+	0.600805i
							\(13\)	−3.22256	+	4.43547i	−0.893777	+	1.23018i	0.0786345	+	0.996904i	\(0.474944\pi\)
−0.972411	+	0.233274i	\(0.925056\pi\)
\(17\)	−1.23763	−	3.93297i	−0.300169	−	0.953886i
							\(19\)	2.51328	−	1.28058i	0.576587	−	0.293786i	−0.141276	−	0.989970i	\(-0.545121\pi\)
0.717863	+	0.696185i	\(0.245121\pi\)
\(23\)	1.87624	−	4.52965i	0.391224	−	0.944498i	−0.598450	−	0.801160i	\(-0.704217\pi\)
							0.989674	−	0.143338i	\(-0.0457835\pi\)
\(29\)	4.07563	+	4.77195i	0.756825	+	0.886128i	0.996402	−	0.0847493i	\(-0.0270089\pi\)
							−0.239578	+	0.970877i	\(0.577009\pi\)
\(31\)	1.39302	−	5.80233i	0.250193	−	1.04213i	−0.695876	−	0.718162i	\(-0.744984\pi\)
							0.946069	−	0.323966i	\(-0.105016\pi\)
\(37\)	1.15845	−	0.989411i	0.190448	−	0.162658i	−0.549111	−	0.835749i	\(-0.685034\pi\)
							0.739560	+	0.673091i	\(0.235034\pi\)
\(41\)	−4.18685	+	4.90218i	−0.653877	+	0.765591i	−0.983966	−	0.178359i	\(-0.942921\pi\)
							0.330089	+	0.943950i	\(0.392921\pi\)
\(43\)	5.04433	−	5.04433i	0.769254	−	0.769254i	−0.208721	−	0.977975i	\(-0.566930\pi\)
							0.977975	+	0.208721i	\(0.0669302\pi\)
\(47\)	−1.66241	−	0.540151i	−0.242488	−	0.0787891i	0.185252	−	0.982691i	\(-0.440690\pi\)
							−0.427740	+	0.903902i	\(0.640690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.15.2	✓	256
11.3	even	5	inner	187.2.r.a.168.2	yes	256
17.8	even	8	inner	187.2.r.a.59.2	yes	256
187.25	even	40	inner	187.2.r.a.25.2	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.15.2	✓	256	1.1	even	1	trivial
187.2.r.a.25.2	yes	256	187.25	even	40	inner
187.2.r.a.59.2	yes	256	17.8	even	8	inner
187.2.r.a.168.2	yes	256	11.3	even	5	inner