Embedded newform 187.2.m.a.142.2

• Label: 187.2.m.a.142.2
• Level: $187$
• Weight: $2$
• Character: 187.142
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.m (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			142.2
Character	\(\chi\)	\(=\)	187.142
Dual form			187.2.m.a.54.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.926579 + 2.23696i) q^{2} +(-1.59986 + 0.318232i) q^{3} +(-2.73122 - 2.73122i) q^{4} +(1.86996 - 2.79859i) q^{5} +(0.770525 - 3.87369i) q^{6} +(2.28141 - 1.52439i) q^{7} +(4.16641 - 1.72579i) q^{8} +(-0.313354 + 0.129795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.926579	+	2.23696i	−0.655190	+	1.58177i	0.149956	+	0.988693i	\(0.452087\pi\)
							−0.805146	+	0.593076i	\(0.797913\pi\)
\(3\)	−1.59986	+	0.318232i	−0.923680	+	0.183731i	−0.633947	−	0.773376i	\(-0.718566\pi\)
							−0.289733	+	0.957108i	\(0.593566\pi\)
\(5\)	1.86996	−	2.79859i	0.836272	−	1.25157i	−0.129345	−	0.991600i	\(-0.541288\pi\)
							0.965617	−	0.259969i	\(-0.0837125\pi\)
\(7\)	2.28141	−	1.52439i	0.862293	−	0.576166i	−0.0438970	−	0.999036i	\(-0.513977\pi\)
							0.906190	+	0.422870i	\(0.138977\pi\)
\(11\)	0.278856	−	3.30488i	0.0840784	−	0.996459i
							\(13\)	−1.29221	−	1.29221i	−0.358394	−	0.358394i	0.504827	−	0.863221i	\(-0.331556\pi\)
−0.863221	+	0.504827i	\(0.831556\pi\)
\(17\)	−2.13859	+	3.52512i	−0.518683	+	0.854966i
							\(19\)	2.18731	−	5.28063i	0.501803	−	1.21146i	−0.446697	−	0.894685i	\(-0.647400\pi\)
0.948501	−	0.316775i	\(-0.102600\pi\)
\(23\)	6.30621	+	1.25438i	1.31493	+	0.261557i	0.802234	−	0.597010i	\(-0.203645\pi\)
							0.512701	+	0.858567i	\(0.328645\pi\)
\(29\)	−3.85019	+	5.76221i	−0.714962	+	1.07002i	0.279000	+	0.960291i	\(0.409997\pi\)
							−0.993962	+	0.109725i	\(0.965003\pi\)
\(31\)	−0.584450	−	2.93823i	−0.104970	−	0.527721i	−0.997111	−	0.0759629i	\(-0.975797\pi\)
							0.892140	−	0.451758i	\(-0.149203\pi\)
\(37\)	3.27986	−	0.652405i	0.539206	−	0.107255i	0.0820281	−	0.996630i	\(-0.473860\pi\)
							0.457178	+	0.889375i	\(0.348860\pi\)
\(41\)	1.57537	−	1.05263i	0.246031	−	0.164392i	−0.426441	−	0.904515i	\(-0.640233\pi\)
							0.672472	+	0.740123i	\(0.265233\pi\)
\(43\)	−2.32497	−	5.61297i	−0.354555	−	0.855970i	−0.996046	−	0.0888401i	\(-0.971684\pi\)
							0.641491	−	0.767130i	\(-0.278316\pi\)
\(47\)	0.752065	−	0.752065i	0.109700	−	0.109700i	−0.650126	−	0.759826i	\(-0.725284\pi\)
							0.759826	+	0.650126i	\(0.225284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.m.a.142.2	yes	128
11.10	odd	2	inner	187.2.m.a.142.15	yes	128
17.3	odd	16	inner	187.2.m.a.54.15	yes	128
187.54	even	16	inner	187.2.m.a.54.2	✓	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.m.a.54.2	✓	128	187.54	even	16	inner
187.2.m.a.54.15	yes	128	17.3	odd	16	inner
187.2.m.a.142.2	yes	128	1.1	even	1	trivial
187.2.m.a.142.15	yes	128	11.10	odd	2	inner