Embedded newform 187.4.g.b.137.17

• Label: 187.4.g.b.137.17
• Level: $187$
• Weight: $4$
• Character: 187.137
• 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			137.17
Character	\(\chi\)	\(=\)	187.137
Dual form			187.4.g.b.86.17

$q$-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 \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.b.86.17	✓	104	11.9	even	5	inner
187.4.g.b.137.17	yes	104	1.1	even	1	trivial
2057.4.a.u.1.19		52	11.8	odd	10
2057.4.a.v.1.34		52	11.3	even	5