Embedded newform 187.2.j.a.16.13

• Label: 187.2.j.a.16.13
• Level: $187$
• Weight: $2$
• Character: 187.16
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			16.13
Character	\(\chi\)	\(=\)	187.16
Dual form			187.2.j.a.152.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.636250 + 1.95818i) q^{2} +(-1.78295 + 2.45402i) q^{3} +(-1.81160 + 1.31621i) q^{4} +(-0.308808 - 0.100338i) q^{5} +(-5.93981 - 1.92996i) q^{6} +(0.911451 + 1.25450i) q^{7} +(-0.398550 - 0.289564i) q^{8} +(-1.91626 - 5.89763i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.636250	+	1.95818i	0.449897	+	1.38464i	0.877023	+	0.480448i	\(0.159526\pi\)
							−0.427127	+	0.904192i	\(0.640474\pi\)
\(3\)	−1.78295	+	2.45402i	−1.02939	+	1.41683i	−0.123981	+	0.992285i	\(0.539566\pi\)
							−0.905407	+	0.424546i	\(0.860434\pi\)
\(5\)	−0.308808	−	0.100338i	−0.138103	−	0.0448725i	0.239150	−	0.970983i	\(-0.423131\pi\)
							−0.377253	+	0.926110i	\(0.623131\pi\)
\(7\)	0.911451	+	1.25450i	0.344496	+	0.474158i	0.945748	−	0.324901i	\(-0.105331\pi\)
							−0.601252	+	0.799060i	\(0.705331\pi\)
\(11\)	2.25299	−	2.43394i	0.679301	−	0.733860i
							\(13\)	0.0304646	+	0.0937603i	0.00844935	+	0.0260044i	0.955192	−	0.295986i	\(-0.0956481\pi\)
−0.946743	+	0.321990i	\(0.895648\pi\)
\(17\)	−1.67766	+	3.76636i	−0.406893	+	0.913476i
							\(19\)	1.43338	+	1.04141i	0.328840	+	0.238916i	0.739938	−	0.672675i	\(-0.234855\pi\)
−0.411098	+	0.911591i	\(0.634855\pi\)
\(23\)			0.138908i			0.0289644i	0.999895	+	0.0144822i	\(0.00460999\pi\)
							−0.999895	+	0.0144822i	\(0.995390\pi\)
\(29\)	−1.70876	−	2.35190i	−0.317308	−	0.436737i	0.620335	−	0.784337i	\(-0.286997\pi\)
							−0.937643	+	0.347600i	\(0.886997\pi\)
\(31\)	4.67478	−	1.51893i	0.839616	−	0.272808i	0.142526	−	0.989791i	\(-0.454478\pi\)
							0.697090	+	0.716983i	\(0.254478\pi\)
\(37\)	4.58765	+	6.31436i	0.754205	+	1.03807i	0.997674	+	0.0681654i	\(0.0217146\pi\)
							−0.243469	+	0.969909i	\(0.578285\pi\)
\(41\)	−4.52331	+	6.22580i	−0.706422	+	0.972306i	0.293445	+	0.955976i	\(0.405198\pi\)
							−0.999867	+	0.0163299i	\(0.994802\pi\)
\(43\)	9.14565			1.39470			0.697349	−	0.716731i	\(-0.254363\pi\)
							0.697349	+	0.716731i	\(0.254363\pi\)
\(47\)	7.64861	+	5.55704i	1.11567	+	0.810578i	0.983546	−	0.180656i	\(-0.0578220\pi\)
							0.132119	+	0.991234i	\(0.457822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.16.13	✓	64
11.9	even	5	inner	187.2.j.a.152.14	yes	64
17.16	even	2	inner	187.2.j.a.16.14	yes	64
187.152	even	10	inner	187.2.j.a.152.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.16.13	✓	64	1.1	even	1	trivial
187.2.j.a.16.14	yes	64	17.16	even	2	inner
187.2.j.a.152.13	yes	64	187.152	even	10	inner
187.2.j.a.152.14	yes	64	11.9	even	5	inner