Embedded newform 187.2.r.a.9.6

• Label: 187.2.r.a.9.6
• Level: $187$
• Weight: $2$
• Character: 187.9
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.r (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			9.6
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.528719 - 1.03767i) q^{2} +(1.63739 - 0.393102i) q^{3} +(0.378355 - 0.520761i) q^{4} +(-3.04927 - 0.239983i) q^{5} +(-1.27363 - 1.49123i) q^{6} +(0.541420 - 2.25518i) q^{7} +(-3.04095 - 0.481640i) q^{8} +(-0.146513 + 0.0746520i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.528719	−	1.03767i	−0.373861	−	0.733744i	0.625041	−	0.780592i	\(-0.285082\pi\)
							−0.998902	+	0.0468483i	\(0.985082\pi\)
\(3\)	1.63739	−	0.393102i	0.945346	−	0.226957i	0.268665	−	0.963234i	\(-0.413418\pi\)
							0.676681	+	0.736276i	\(0.263418\pi\)
\(5\)	−3.04927	−	0.239983i	−1.36368	−	0.107324i	−0.624600	−	0.780945i	\(-0.714738\pi\)
							−0.739077	+	0.673621i	\(0.764738\pi\)
\(7\)	0.541420	−	2.25518i	0.204637	−	0.852376i	−0.771676	−	0.636016i	\(-0.780581\pi\)
							0.976314	−	0.216361i	\(-0.0694187\pi\)
\(11\)	−0.243128	−	3.30770i	−0.0733058	−	0.997310i
							\(13\)	6.24182	+	2.02809i	1.73117	+	0.562491i	0.993618	−	0.112799i	\(-0.0359818\pi\)
0.737552	+	0.675291i	\(0.235982\pi\)
\(17\)	3.22674	+	2.56674i	0.782599	+	0.622526i
							\(19\)	0.0786539	−	0.496601i	0.0180444	−	0.113928i	−0.977022	−	0.213138i	\(-0.931632\pi\)
0.995067	+	0.0992100i	\(0.0316316\pi\)
\(23\)	2.77259	−	1.14845i	0.578126	−	0.239468i	−0.0744070	−	0.997228i	\(-0.523706\pi\)
							0.652533	+	0.757760i	\(0.273706\pi\)
\(29\)	2.96854	−	1.81912i	0.551244	−	0.337803i	−0.218824	−	0.975764i	\(-0.570222\pi\)
							0.770068	+	0.637961i	\(0.220222\pi\)
\(31\)	−2.98316	+	3.49284i	−0.535792	+	0.627332i	−0.960594	−	0.277954i	\(-0.910344\pi\)
							0.424802	+	0.905286i	\(0.360344\pi\)
\(37\)	−2.87520	−	4.69190i	−0.472680	−	0.771343i	0.524089	−	0.851663i	\(-0.324406\pi\)
							−0.996769	+	0.0803201i	\(0.974406\pi\)
\(41\)	9.98634	+	6.11964i	1.55960	+	0.955727i	0.991111	+	0.133039i	\(0.0424735\pi\)
							0.568493	+	0.822688i	\(0.307526\pi\)
\(43\)	1.65005	+	1.65005i	0.251630	+	0.251630i	0.821639	−	0.570009i	\(-0.193060\pi\)
							−0.570009	+	0.821639i	\(0.693060\pi\)
\(47\)	−3.55540	−	4.89359i	−0.518608	−	0.713803i	0.466733	−	0.884398i	\(-0.345431\pi\)
							−0.985341	+	0.170595i	\(0.945431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.6	✓	256
11.5	even	5	inner	187.2.r.a.60.11	yes	256
17.2	even	8	inner	187.2.r.a.53.11	yes	256
187.104	even	40	inner	187.2.r.a.104.6	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.6	✓	256	1.1	even	1	trivial
187.2.r.a.53.11	yes	256	17.2	even	8	inner
187.2.r.a.60.11	yes	256	11.5	even	5	inner
187.2.r.a.104.6	yes	256	187.104	even	40	inner