Embedded newform 187.2.r.a.9.13

• Label: 187.2.r.a.9.13
• Level: $187$
• Weight: $2$
• Character: 187.9
• 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			9.13
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.808667 + 1.58710i) q^{2} +(-1.31588 + 0.315915i) q^{3} +(-0.689370 + 0.948836i) q^{4} +(3.89226 + 0.306327i) q^{5} +(-1.56550 - 1.83296i) q^{6} +(-0.115270 + 0.480136i) q^{7} +(1.45526 + 0.230490i) q^{8} +(-1.04128 + 0.530561i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.808667	+	1.58710i	0.571814	+	1.12225i	0.978029	+	0.208471i	\(0.0668486\pi\)
							−0.406214	+	0.913778i	\(0.633151\pi\)
\(3\)	−1.31588	+	0.315915i	−0.759723	+	0.182393i	−0.594757	−	0.803906i	\(-0.702752\pi\)
							−0.164966	+	0.986299i	\(0.552752\pi\)
\(5\)	3.89226	+	0.306327i	1.74067	+	0.136994i	0.908598	−	0.417672i	\(-0.137154\pi\)
							0.832073	+	0.554666i	\(0.187154\pi\)
\(7\)	−0.115270	+	0.480136i	−0.0435681	+	0.181474i	−0.989911	−	0.141690i	\(-0.954747\pi\)
							0.946343	+	0.323164i	\(0.104747\pi\)
\(11\)	−3.28875	−	0.429086i	−0.991596	−	0.129374i
							\(13\)	−0.891998	−	0.289828i	−0.247396	−	0.0803837i	0.182694	−	0.983170i	\(-0.441518\pi\)
−0.430090	+	0.902786i	\(0.641518\pi\)
\(17\)	−0.396461	−	4.10400i	−0.0961558	−	0.995366i
							\(19\)	0.0652483	−	0.411962i	0.0149690	−	0.0945105i	−0.979073	−	0.203511i	\(-0.934765\pi\)
0.994042	+	0.109000i	\(0.0347649\pi\)
\(23\)	−2.69425	+	1.11600i	−0.561790	+	0.232701i	−0.645462	−	0.763792i	\(-0.723335\pi\)
							0.0836720	+	0.996493i	\(0.473335\pi\)
\(29\)	4.13343	−	2.53297i	0.767558	−	0.470360i	−0.0827991	−	0.996566i	\(-0.526386\pi\)
							0.850357	+	0.526206i	\(0.176386\pi\)
\(31\)	0.496740	−	0.581607i	0.0892171	−	0.104460i	−0.714002	−	0.700144i	\(-0.753119\pi\)
							0.803219	+	0.595684i	\(0.203119\pi\)
\(37\)	−3.08708	−	5.03766i	−0.507513	−	0.828186i	0.491577	−	0.870834i	\(-0.336421\pi\)
							−0.999090	+	0.0426483i	\(0.986421\pi\)
\(41\)	−7.12177	−	4.36423i	−1.11223	−	0.681578i	−0.160051	−	0.987109i	\(-0.551166\pi\)
							−0.952182	+	0.305531i	\(0.901166\pi\)
\(43\)	−2.84118	−	2.84118i	−0.433276	−	0.433276i	0.456465	−	0.889741i	\(-0.349115\pi\)
							−0.889741	+	0.456465i	\(0.849115\pi\)
\(47\)	−2.56127	−	3.52529i	−0.373600	−	0.514216i	0.580275	−	0.814421i	\(-0.302945\pi\)
							−0.953875	+	0.300204i	\(0.902945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.13	✓	256
11.5	even	5	inner	187.2.r.a.60.4	yes	256
17.2	even	8	inner	187.2.r.a.53.4	yes	256
187.104	even	40	inner	187.2.r.a.104.13	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.13	✓	256	1.1	even	1	trivial
187.2.r.a.53.4	yes	256	17.2	even	8	inner
187.2.r.a.60.4	yes	256	11.5	even	5	inner
187.2.r.a.104.13	yes	256	187.104	even	40	inner