Embedded newform 287.2.l.a.27.9

• Label: 287.2.l.a.27.9
• Level: $287$
• Weight: $2$
• Character: 287.27
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			27.9
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.717741 - 0.717741i) q^{2} +(-0.472410 - 1.14050i) q^{3} -0.969695i q^{4} +(-1.70830 - 1.70830i) q^{5} +(-0.479514 + 1.15765i) q^{6} +(-2.54594 - 0.719863i) q^{7} +(-2.13147 + 2.13147i) q^{8} +(1.04376 - 1.04376i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.717741	−	0.717741i	−0.507520	−	0.507520i	0.406245	−	0.913764i	\(-0.366838\pi\)
							−0.913764	+	0.406245i	\(0.866838\pi\)
\(3\)	−0.472410	−	1.14050i	−0.272746	−	0.658467i	0.726853	−	0.686793i	\(-0.240982\pi\)
							−0.999599	+	0.0283265i	\(0.990982\pi\)
\(5\)	−1.70830	−	1.70830i	−0.763977	−	0.763977i	0.213062	−	0.977039i	\(-0.431656\pi\)
							−0.977039	+	0.213062i	\(0.931656\pi\)
\(7\)	−2.54594	−	0.719863i	−0.962274	−	0.272083i
							\(11\)	1.64933	+	3.98183i	0.497291	+	1.20057i	0.950937	+	0.309385i	\(0.100123\pi\)
−0.453646	+	0.891182i	\(0.649877\pi\)
\(13\)	0.242554	+	0.585577i	0.0672723	+	0.162410i	0.953940	−	0.299997i	\(-0.0969858\pi\)
							−0.886668	+	0.462407i	\(0.846986\pi\)
\(17\)	−0.404436	+	0.976394i	−0.0980900	+	0.236810i	−0.965306	−	0.261122i	\(-0.915907\pi\)
							0.867216	+	0.497933i	\(0.165907\pi\)
\(19\)	1.33325	−	3.21875i	0.305868	−	0.738431i	−0.693962	−	0.720012i	\(-0.744136\pi\)
							0.999830	−	0.0184197i	\(-0.00586350\pi\)
\(23\)		−	0.864061i		−	0.180169i	−0.995934	−	0.0900846i	\(-0.971286\pi\)
							0.995934	−	0.0900846i	\(-0.0287137\pi\)
\(29\)	3.31396	−	1.37269i	0.615387	−	0.254902i	−0.0531429	−	0.998587i	\(-0.516924\pi\)
							0.668530	+	0.743685i	\(0.266924\pi\)
\(31\)	−10.4379			−1.87470			−0.937348	−	0.348394i	\(-0.886727\pi\)
							−0.937348	+	0.348394i	\(0.886727\pi\)
\(37\)	−8.32296			−1.36829			−0.684143	−	0.729348i	\(-0.739824\pi\)
							−0.684143	+	0.729348i	\(0.739824\pi\)
\(41\)	−4.90802	+	4.11234i	−0.766503	+	0.642240i
							\(43\)	−3.79548	−	3.79548i	−0.578805	−	0.578805i	0.355769	−	0.934574i	\(-0.384219\pi\)
−0.934574	+	0.355769i	\(0.884219\pi\)
\(47\)	2.19038	−	5.28804i	0.319500	−	0.771340i	−0.679781	−	0.733415i	\(-0.737925\pi\)
							0.999281	−	0.0379251i	\(-0.0120748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.l.a.27.9	✓	104
7.6	odd	2	inner	287.2.l.a.27.10	yes	104
41.38	odd	8	inner	287.2.l.a.202.10	yes	104
287.202	even	8	inner	287.2.l.a.202.9	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.l.a.27.9	✓	104	1.1	even	1	trivial
287.2.l.a.27.10	yes	104	7.6	odd	2	inner
287.2.l.a.202.9	yes	104	287.202	even	8	inner
287.2.l.a.202.10	yes	104	41.38	odd	8	inner