Embedded newform 187.4.g.b.86.17

• Label: 187.4.g.b.86.17
• Level: $187$
• Weight: $4$
• Character: 187.86
• Analytic conductor: $11.033$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	187.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0333571711\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			86.17
Character	\(\chi\)	\(=\)	187.86
Dual form			187.4.g.b.137.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.670374 - 2.06320i) q^{2} +(5.26729 - 3.82691i) q^{3} +(2.66475 + 1.93605i) q^{4} +(-3.37323 - 10.3817i) q^{5} +(-4.36462 - 13.4329i) q^{6} +(-20.4591 - 14.8644i) q^{7} +(19.8213 - 14.4010i) q^{8} +(4.75564 - 14.6364i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q + 2 q^{2} - 10 q^{3} - 130 q^{4} - 50 q^{5} + 12 q^{6} + 36 q^{7} - 220 q^{8} - 286 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.670374	−	2.06320i	0.237013	−	0.729451i	−0.759835	−	0.650116i	\(-0.774720\pi\)
							0.996848	−	0.0793352i	\(-0.0252797\pi\)
\(3\)	5.26729	−	3.82691i	1.01369	−	0.736489i	0.0487105	−	0.998813i	\(-0.484489\pi\)
							0.964980	+	0.262324i	\(0.0844888\pi\)
\(5\)	−3.37323	−	10.3817i	−0.301710	−	0.928569i	−0.980884	−	0.194592i	\(-0.937662\pi\)
							0.679174	−	0.733977i	\(-0.262338\pi\)
\(7\)	−20.4591	−	14.8644i	−1.10469	−	0.802602i	−0.122868	−	0.992423i	\(-0.539209\pi\)
							−0.981819	+	0.189821i	\(0.939209\pi\)
\(11\)	−35.8590	+	6.71817i	−0.982899	+	0.184146i
							\(13\)	−9.57460	+	29.4676i	−0.204271	+	0.628680i	0.795472	+	0.605990i	\(0.207223\pi\)
−0.999743	+	0.0226898i	\(0.992777\pi\)
\(17\)	−5.25329	−	16.1680i	−0.0749476	−	0.230665i
							\(19\)	117.774	−	85.5677i	1.42206	−	1.03319i	0.430635	−	0.902526i	\(-0.358290\pi\)
0.991427	−	0.130662i	\(-0.0417102\pi\)
\(23\)	94.4647			0.856402			0.428201	−	0.903683i	\(-0.359148\pi\)
							0.428201	+	0.903683i	\(0.359148\pi\)
\(29\)	19.9170	+	14.4705i	0.127534	+	0.0926589i	0.649724	−	0.760171i	\(-0.274885\pi\)
							−0.522190	+	0.852829i	\(0.674885\pi\)
\(31\)	−13.3321	+	41.0320i	−0.0772425	+	0.237728i	−0.982221	−	0.187730i	\(-0.939887\pi\)
							0.904978	+	0.425458i	\(0.139887\pi\)
\(37\)	−67.8240	−	49.2770i	−0.301357	−	0.218949i	0.426822	−	0.904336i	\(-0.359633\pi\)
							−0.728179	+	0.685387i	\(0.759633\pi\)
\(41\)	361.288	−	262.491i	1.37619	−	0.999858i	0.378961	−	0.925412i	\(-0.376281\pi\)
							0.997225	−	0.0744456i	\(-0.0237187\pi\)
\(43\)	−489.756			−1.73691			−0.868455	−	0.495768i	\(-0.834887\pi\)
							−0.868455	+	0.495768i	\(0.834887\pi\)
\(47\)	201.957	−	146.730i	0.626775	−	0.455378i	−0.228507	−	0.973542i	\(-0.573384\pi\)
							0.855281	+	0.518164i	\(0.173384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.4.g.b.86.17	✓	104
11.4	even	5		2057.4.a.v.1.34		52
11.5	even	5	inner	187.4.g.b.137.17	yes	104
11.7	odd	10		2057.4.a.u.1.19		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.b.86.17	✓	104	1.1	even	1	trivial
187.4.g.b.137.17	yes	104	11.5	even	5	inner
2057.4.a.u.1.19		52	11.7	odd	10
2057.4.a.v.1.34		52	11.4	even	5