Embedded newform 287.2.j.a.81.16

• Label: 287.2.j.a.81.16
• Level: $287$
• Weight: $2$
• Character: 287.81
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			81.16
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0647109 + 0.112083i) q^{2} +(1.92881 + 1.11360i) q^{3} +(0.991625 - 1.71754i) q^{4} +(0.451146 + 0.781409i) q^{5} +0.288247i q^{6} +(-2.43969 + 1.02368i) q^{7} +0.515519 q^{8} +(0.980198 + 1.69775i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0647109	+	0.112083i	0.0457575	+	0.0792543i	0.887997	−	0.459849i	\(-0.152097\pi\)
							−0.842240	+	0.539103i	\(0.818763\pi\)
\(3\)	1.92881	+	1.11360i	1.11360	+	0.642936i	0.939759	−	0.341838i	\(-0.111050\pi\)
							0.173839	+	0.984774i	\(0.444383\pi\)
\(5\)	0.451146	+	0.781409i	0.201759	+	0.349457i	0.949095	−	0.314989i	\(-0.102001\pi\)
							−0.747336	+	0.664446i	\(0.768668\pi\)
\(7\)	−2.43969	+	1.02368i	−0.922116	+	0.386913i
							\(11\)	4.20583	+	2.42823i	1.26810	+	0.732140i	0.974629	−	0.223826i	\(-0.0718548\pi\)
0.293475	+	0.955967i	\(0.405188\pi\)
\(13\)		−	1.67851i		−	0.465535i	−0.972532	−	0.232768i	\(-0.925222\pi\)
							0.972532	−	0.232768i	\(-0.0747782\pi\)
\(17\)	1.41711	+	0.818170i	0.343700	+	0.198435i	0.661907	−	0.749586i	\(-0.269747\pi\)
							−0.318207	+	0.948021i	\(0.603081\pi\)
\(19\)	−4.89577	+	2.82658i	−1.12317	+	0.648461i	−0.942208	−	0.335030i	\(-0.891254\pi\)
							−0.180960	+	0.983491i	\(0.557920\pi\)
\(23\)	−4.04340	−	7.00337i	−0.843106	−	1.46030i	−0.887256	−	0.461278i	\(-0.847391\pi\)
							0.0441490	−	0.999025i	\(-0.485942\pi\)
\(29\)			0.161060i			0.0299081i	0.999888	+	0.0149541i	\(0.00476020\pi\)
							−0.999888	+	0.0149541i	\(0.995240\pi\)
\(31\)	−3.61898	+	6.26826i	−0.649988	+	1.12581i	0.333137	+	0.942878i	\(0.391893\pi\)
							−0.983125	+	0.182934i	\(0.941440\pi\)
\(37\)	4.12698	+	7.14814i	0.678471	+	1.17515i	0.975441	+	0.220260i	\(0.0706906\pi\)
							−0.296970	+	0.954887i	\(0.595976\pi\)
\(41\)	−5.72162	−	2.87455i	−0.893567	−	0.448929i
							\(43\)	2.44646			0.373081			0.186541	−	0.982447i	\(-0.440272\pi\)
0.186541	+	0.982447i	\(0.440272\pi\)
\(47\)	−7.05669	+	4.07418i	−1.02932	+	0.594281i	−0.916791	−	0.399367i	\(-0.869230\pi\)
							−0.112533	+	0.993648i	\(0.535896\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.16	yes	52
7.2	even	3	inner	287.2.j.a.163.15	yes	52
41.40	even	2	inner	287.2.j.a.81.15	✓	52
287.163	even	6	inner	287.2.j.a.163.16	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.15	✓	52	41.40	even	2	inner
287.2.j.a.81.16	yes	52	1.1	even	1	trivial
287.2.j.a.163.15	yes	52	7.2	even	3	inner
287.2.j.a.163.16	yes	52	287.163	even	6	inner