Embedded newform 187.2.j.a.16.9

• Label: 187.2.j.a.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	187.16
Dual form			187.2.j.a.152.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0862831 + 0.265552i) q^{2} +(-1.41541 + 1.94814i) q^{3} +(1.55496 - 1.12975i) q^{4} +(3.81388 + 1.23921i) q^{5} +(-0.639458 - 0.207772i) q^{6} +(-2.14946 - 2.95848i) q^{7} +(0.885957 + 0.643686i) q^{8} +(-0.864821 - 2.66164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0862831	+	0.265552i	0.0610114	+	0.187774i	0.976917	−	0.213621i	\(-0.0685257\pi\)
							−0.915905	+	0.401394i	\(0.868526\pi\)
\(3\)	−1.41541	+	1.94814i	−0.817185	+	1.12476i	0.172990	+	0.984924i	\(0.444657\pi\)
							−0.990175	+	0.139835i	\(0.955343\pi\)
\(5\)	3.81388	+	1.23921i	1.70562	+	0.554189i	0.989594	−	0.143885i	\(-0.0459596\pi\)
							0.716025	+	0.698074i	\(0.245960\pi\)
\(7\)	−2.14946	−	2.95848i	−0.812419	−	1.11820i	−0.990946	−	0.134263i	\(-0.957133\pi\)
							0.178527	−	0.983935i	\(-0.442867\pi\)
\(11\)	0.968616	+	3.17203i	0.292049	+	0.956403i
							\(13\)	0.396206	+	1.21940i	0.109888	+	0.338200i	0.990847	−	0.134993i	\(-0.0431013\pi\)
−0.880959	+	0.473193i	\(0.843101\pi\)
\(17\)	−3.07041	+	2.75183i	−0.744685	+	0.667416i
							\(19\)	−2.75738	−	2.00335i	−0.632586	−	0.459601i	0.224709	−	0.974426i	\(-0.427857\pi\)
−0.857295	+	0.514825i	\(0.827857\pi\)
\(23\)		−	1.45345i		−	0.303065i	−0.988452	−	0.151532i	\(-0.951579\pi\)
							0.988452	−	0.151532i	\(-0.0484207\pi\)
\(29\)	−2.97119	−	4.08949i	−0.551736	−	0.759399i	0.438511	−	0.898726i	\(-0.355506\pi\)
							−0.990246	+	0.139327i	\(0.955506\pi\)
\(31\)	2.26878	−	0.737170i	0.407484	−	0.132400i	−0.0981007	−	0.995176i	\(-0.531277\pi\)
							0.505585	+	0.862777i	\(0.331277\pi\)
\(37\)	−3.80706	−	5.23997i	−0.625878	−	0.861447i	0.371887	−	0.928278i	\(-0.378711\pi\)
							−0.997764	+	0.0668316i	\(0.978711\pi\)
\(41\)	2.93999	−	4.04655i	0.459149	−	0.631964i	−0.515183	−	0.857080i	\(-0.672276\pi\)
							0.974332	+	0.225116i	\(0.0722760\pi\)
\(43\)	−6.03663			−0.920577			−0.460288	−	0.887769i	\(-0.652254\pi\)
							−0.460288	+	0.887769i	\(0.652254\pi\)
\(47\)	4.42169	+	3.21255i	0.644970	+	0.468598i	0.861554	−	0.507666i	\(-0.169492\pi\)
							−0.216584	+	0.976264i	\(0.569492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.16.9	✓	64
11.9	even	5	inner	187.2.j.a.152.10	yes	64
17.16	even	2	inner	187.2.j.a.16.10	yes	64
187.152	even	10	inner	187.2.j.a.152.9	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.16.9	✓	64	1.1	even	1	trivial
187.2.j.a.16.10	yes	64	17.16	even	2	inner
187.2.j.a.152.9	yes	64	187.152	even	10	inner
187.2.j.a.152.10	yes	64	11.9	even	5	inner