Embedded newform 287.2.j.a.81.2

• Label: 287.2.j.a.81.2
• 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			81.2
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30798 - 2.26549i) q^{2} +(0.306987 + 0.177239i) q^{3} +(-2.42163 + 4.19439i) q^{4} +(0.326729 + 0.565911i) q^{5} -0.927301i q^{6} +(-0.262924 - 2.63265i) q^{7} +7.43789 q^{8} +(-1.43717 - 2.48926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30798	−	2.26549i	−0.924883	−	1.60194i	−0.791749	−	0.610846i	\(-0.790829\pi\)
							−0.133134	−	0.991098i	\(-0.542504\pi\)
\(3\)	0.306987	+	0.177239i	0.177239	+	0.102329i	0.585995	−	0.810315i	\(-0.300704\pi\)
							−0.408756	+	0.912644i	\(0.634037\pi\)
\(5\)	0.326729	+	0.565911i	0.146118	+	0.253083i	0.929789	−	0.368092i	\(-0.119989\pi\)
							−0.783672	+	0.621175i	\(0.786656\pi\)
\(7\)	−0.262924	−	2.63265i	−0.0993758	−	0.995050i
							\(11\)	−4.27447	−	2.46787i	−1.28880	−	0.744090i	−0.310362	−	0.950619i	\(-0.600450\pi\)
−0.978441	+	0.206528i	\(0.933783\pi\)
\(13\)			3.57652i			0.991948i	0.868337	+	0.495974i	\(0.165189\pi\)
							−0.868337	+	0.495974i	\(0.834811\pi\)
\(17\)	−3.73134	−	2.15429i	−0.904983	−	0.522492i	−0.0261696	−	0.999658i	\(-0.508331\pi\)
							−0.878814	+	0.477165i	\(0.841664\pi\)
\(19\)	−3.55801	+	2.05422i	−0.816264	+	0.471270i	−0.849127	−	0.528190i	\(-0.822871\pi\)
							0.0328623	+	0.999460i	\(0.489538\pi\)
\(23\)	0.237698	+	0.411705i	0.0495634	+	0.0858464i	0.889743	−	0.456462i	\(-0.150884\pi\)
							−0.840179	+	0.542309i	\(0.817550\pi\)
\(29\)		−	6.69956i		−	1.24408i	−0.782987	−	0.622038i	\(-0.786305\pi\)
							0.782987	−	0.622038i	\(-0.213695\pi\)
\(31\)	−2.78540	+	4.82445i	−0.500272	+	0.866497i	0.499728	+	0.866183i	\(0.333433\pi\)
							−1.00000		0.000314414i	\(0.999900\pi\)
\(37\)	−0.927079	−	1.60575i	−0.152411	−	0.263983i	0.779702	−	0.626150i	\(-0.215370\pi\)
							−0.932113	+	0.362167i	\(0.882037\pi\)
\(41\)	6.11107	−	1.91177i	0.954388	−	0.298569i
							\(43\)	−0.240149			−0.0366224			−0.0183112	−	0.999832i	\(-0.505829\pi\)
−0.0183112	+	0.999832i	\(0.505829\pi\)
\(47\)	−9.69952	+	5.60002i	−1.41482	+	0.816847i	−0.995837	−	0.0911465i	\(-0.970947\pi\)
							−0.418984	+	0.907994i	\(0.637614\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.2	yes	52
7.2	even	3	inner	287.2.j.a.163.1	yes	52
41.40	even	2	inner	287.2.j.a.81.1	✓	52
287.163	even	6	inner	287.2.j.a.163.2	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.1	✓	52	41.40	even	2	inner
287.2.j.a.81.2	yes	52	1.1	even	1	trivial
287.2.j.a.163.1	yes	52	7.2	even	3	inner
287.2.j.a.163.2	yes	52	287.163	even	6	inner