Embedded newform 187.2.m.a.10.5

• Label: 187.2.m.a.10.5
• Level: $187$
• Weight: $2$
• Character: 187.10
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• 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			10.5
Character	\(\chi\)	\(=\)	187.10
Dual form			187.2.m.a.131.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30126 - 0.539000i) q^{2} +(-2.38581 - 1.59415i) q^{3} +(-0.0114548 - 0.0114548i) q^{4} +(-3.07041 + 0.610742i) q^{5} +(2.24531 + 3.36035i) q^{6} +(0.785949 - 3.95123i) q^{7} +(1.08673 + 2.62360i) q^{8} +(2.00273 + 4.83502i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30126	−	0.539000i	−0.920130	−	0.381130i	−0.128204	−	0.991748i	\(-0.540921\pi\)
							−0.791926	+	0.610617i	\(0.790921\pi\)
\(3\)	−2.38581	−	1.59415i	−1.37745	−	0.920381i	−0.377463	−	0.926025i	\(-0.623203\pi\)
							−0.999984	+	0.00564401i	\(0.998203\pi\)
\(5\)	−3.07041	+	0.610742i	−1.37313	+	0.273132i	−0.825883	−	0.563841i	\(-0.809323\pi\)
							−0.547244	+	0.836973i	\(0.684323\pi\)
\(7\)	0.785949	−	3.95123i	0.297061	−	1.49343i	−0.487363	−	0.873200i	\(-0.662041\pi\)
							0.784423	−	0.620226i	\(-0.212959\pi\)
\(11\)	−2.90472	−	1.60082i	−0.875805	−	0.482665i
							\(13\)	2.62889	+	2.62889i	0.729124	+	0.729124i	0.970445	−	0.241321i	\(-0.0775808\pi\)
−0.241321	+	0.970445i	\(0.577581\pi\)
\(17\)	3.70164	+	1.81600i	0.897779	+	0.440445i
							\(19\)	−3.36364	−	1.39327i	−0.771673	−	0.319637i	−0.0381232	−	0.999273i	\(-0.512138\pi\)
−0.733550	+	0.679636i	\(0.762138\pi\)
\(23\)	0.347005	−	0.231861i	0.0723556	−	0.0483464i	−0.518866	−	0.854855i	\(-0.673646\pi\)
							0.591222	+	0.806509i	\(0.298646\pi\)
\(29\)	−3.98923	+	0.793506i	−0.740781	+	0.147350i	−0.551034	−	0.834483i	\(-0.685767\pi\)
							−0.189746	+	0.981833i	\(0.560767\pi\)
\(31\)	−4.24988	+	6.36039i	−0.763301	+	1.14236i	0.222580	+	0.974914i	\(0.428552\pi\)
							−0.985881	+	0.167446i	\(0.946448\pi\)
\(37\)	−4.99821	−	3.33970i	−0.821701	−	0.549043i	0.0721569	−	0.997393i	\(-0.477012\pi\)
							−0.893858	+	0.448350i	\(0.852012\pi\)
\(41\)	−0.955894	+	4.80561i	−0.149286	+	0.750509i	0.831516	+	0.555501i	\(0.187473\pi\)
							−0.980802	+	0.195009i	\(0.937527\pi\)
\(43\)	4.62394	−	1.91530i	0.705144	−	0.292080i	−0.00114951	−	0.999999i	\(-0.500366\pi\)
							0.706293	+	0.707919i	\(0.250366\pi\)
\(47\)	0.913702	−	0.913702i	0.133277	−	0.133277i	−0.637321	−	0.770598i	\(-0.719958\pi\)
							0.770598	+	0.637321i	\(0.219958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.m.a.10.5	✓	128
11.10	odd	2	inner	187.2.m.a.10.12	yes	128
17.12	odd	16	inner	187.2.m.a.131.12	yes	128
187.131	even	16	inner	187.2.m.a.131.5	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.m.a.10.5	✓	128	1.1	even	1	trivial
187.2.m.a.10.12	yes	128	11.10	odd	2	inner
187.2.m.a.131.5	yes	128	187.131	even	16	inner
187.2.m.a.131.12	yes	128	17.12	odd	16	inner