Embedded newform 187.2.t.a.6.3

• Label: 187.2.t.a.6.3
• Level: $187$
• Weight: $2$
• Character: 187.6
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.t (of order \(80\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(16\) over \(\Q(\zeta_{80})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{80}]$

Embedding invariants

Embedding label			6.3
Character	\(\chi\)	\(=\)	187.6
Dual form			187.2.t.a.156.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.09150 - 0.164604i) q^{2} +(0.560652 + 0.258464i) q^{3} +(2.37189 + 0.375671i) q^{4} +(-1.69373 - 2.14848i) q^{5} +(-1.13006 - 0.632864i) q^{6} +(-0.674528 + 1.82839i) q^{7} +(-0.818977 - 0.196619i) q^{8} +(-1.70082 - 1.99140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 40 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 40 q^{6} - 40 q^{7} - 40 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.09150	−	0.164604i	−1.47891	−	0.116393i	−0.686835	−	0.726813i	\(-0.741001\pi\)
							−0.792078	+	0.610420i	\(0.791001\pi\)
\(3\)	0.560652	+	0.258464i	0.323693	+	0.149224i	0.573804	−	0.818993i	\(-0.305467\pi\)
							−0.250111	+	0.968217i	\(0.580467\pi\)
\(5\)	−1.69373	−	2.14848i	−0.757459	−	0.960832i	0.242498	−	0.970152i	\(-0.422033\pi\)
							−0.999957	+	0.00932028i	\(0.997033\pi\)
\(7\)	−0.674528	+	1.82839i	−0.254948	+	0.691066i	0.744770	+	0.667321i	\(0.232559\pi\)
							−0.999718	+	0.0237452i	\(0.992441\pi\)
\(11\)	−3.05692	+	1.28656i	−0.921696	+	0.387912i
							\(13\)	0.445105	−	0.873568i	0.123450	−	0.242284i	−0.821008	−	0.570917i	\(-0.806588\pi\)
0.944458	+	0.328633i	\(0.106588\pi\)
\(17\)	−0.662545	−	4.06953i	−0.160691	−	0.987005i
							\(19\)	−3.67349	+	2.25112i	−0.842758	+	0.516442i	−0.875611	−	0.483018i	\(-0.839541\pi\)
0.0328530	+	0.999460i	\(0.489541\pi\)
\(23\)	−9.16567	−	1.82317i	−1.91117	−	0.380156i	−0.911650	−	0.410968i	\(-0.865191\pi\)
							−0.999525	+	0.0308117i	\(0.990191\pi\)
\(29\)	−0.00510960	−	0.130048i	−0.000948829	−	0.0241493i	0.998682	−	0.0513316i	\(-0.0163465\pi\)
							−0.999630	+	0.0271823i	\(0.991347\pi\)
\(31\)	−3.46290	+	1.93931i	−0.621954	+	0.348311i	−0.767508	−	0.641039i	\(-0.778504\pi\)
							0.145554	+	0.989350i	\(0.453504\pi\)
\(37\)	6.27840	−	6.79193i	1.03216	−	1.11659i	0.0390717	−	0.999236i	\(-0.487560\pi\)
							0.993091	−	0.117350i	\(-0.0374401\pi\)
\(41\)	5.02060	+	0.197260i	0.784086	+	0.0308068i	0.428155	−	0.903705i	\(-0.359164\pi\)
							0.355931	+	0.934512i	\(0.384164\pi\)
\(43\)	3.05882	+	7.38466i	0.466466	+	1.12615i	0.965695	+	0.259679i	\(0.0836168\pi\)
							−0.499229	+	0.866470i	\(0.666383\pi\)
\(47\)	4.28534	−	0.678731i	0.625081	−	0.0990031i	0.164145	−	0.986436i	\(-0.447514\pi\)
							0.460936	+	0.887433i	\(0.347514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.t.a.6.3	✓	512
11.2	odd	10	inner	187.2.t.a.57.3	yes	512
17.3	odd	16	inner	187.2.t.a.105.3	yes	512
187.156	even	80	inner	187.2.t.a.156.3	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.t.a.6.3	✓	512	1.1	even	1	trivial
187.2.t.a.57.3	yes	512	11.2	odd	10	inner
187.2.t.a.105.3	yes	512	17.3	odd	16	inner
187.2.t.a.156.3	yes	512	187.156	even	80	inner