Embedded newform 287.2.l.a.27.16

• Label: 287.2.l.a.27.16
• 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			27.16
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.279520 + 0.279520i) q^{2} +(0.739386 + 1.78504i) q^{3} -1.84374i q^{4} +(-2.85060 - 2.85060i) q^{5} +(-0.292280 + 0.705627i) q^{6} +(-0.406011 - 2.61441i) q^{7} +(1.07440 - 1.07440i) q^{8} +(-0.518341 + 0.518341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.279520	+	0.279520i	0.197651	+	0.197651i	0.798992	−	0.601341i	\(-0.205367\pi\)
							−0.601341	+	0.798992i	\(0.705367\pi\)
\(3\)	0.739386	+	1.78504i	0.426885	+	1.03059i	0.980270	+	0.197665i	\(0.0633359\pi\)
							−0.553385	+	0.832926i	\(0.686664\pi\)
\(5\)	−2.85060	−	2.85060i	−1.27483	−	1.27483i	−0.943524	−	0.331305i	\(-0.892511\pi\)
							−0.331305	−	0.943524i	\(-0.607489\pi\)
\(7\)	−0.406011	−	2.61441i	−0.153458	−	0.988155i
							\(11\)	−0.392164	−	0.946767i	−0.118242	−	0.285461i	0.853667	−	0.520820i	\(-0.174374\pi\)
−0.971909	+	0.235358i	\(0.924374\pi\)
\(13\)	0.529743	+	1.27891i	0.146924	+	0.354707i	0.980159	−	0.198213i	\(-0.0635137\pi\)
							−0.833235	+	0.552919i	\(0.813514\pi\)
\(17\)	1.48320	−	3.58077i	0.359729	−	0.868463i	−0.635608	−	0.772012i	\(-0.719251\pi\)
							0.995338	−	0.0964516i	\(-0.0307493\pi\)
\(19\)	−2.65648	+	6.41330i	−0.609438	+	1.47131i	0.254176	+	0.967158i	\(0.418196\pi\)
							−0.863614	+	0.504154i	\(0.831804\pi\)
\(23\)			0.606245i			0.126411i	0.998001	+	0.0632054i	\(0.0201323\pi\)
							−0.998001	+	0.0632054i	\(0.979868\pi\)
\(29\)	8.99131	−	3.72432i	1.66964	−	0.691589i	0.670893	−	0.741555i	\(-0.265911\pi\)
							0.998751	+	0.0499656i	\(0.0159112\pi\)
\(31\)	7.26125			1.30416			0.652079	−	0.758151i	\(-0.273897\pi\)
							0.652079	+	0.758151i	\(0.273897\pi\)
\(37\)	−8.67023			−1.42538			−0.712689	−	0.701481i	\(-0.752523\pi\)
							−0.712689	+	0.701481i	\(0.752523\pi\)
\(41\)	0.361720	−	6.39290i	0.0564912	−	0.998403i
							\(43\)	2.62048	+	2.62048i	0.399620	+	0.399620i	0.878099	−	0.478479i	\(-0.158812\pi\)
−0.478479	+	0.878099i	\(0.658812\pi\)
\(47\)	1.95784	−	4.72664i	0.285580	−	0.689451i	−0.714367	−	0.699771i	\(-0.753285\pi\)
							0.999947	+	0.0103206i	\(0.00328520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.l.a.27.16	yes	104
7.6	odd	2	inner	287.2.l.a.27.15	✓	104
41.38	odd	8	inner	287.2.l.a.202.15	yes	104
287.202	even	8	inner	287.2.l.a.202.16	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.l.a.27.15	✓	104	7.6	odd	2	inner
287.2.l.a.27.16	yes	104	1.1	even	1	trivial
287.2.l.a.202.15	yes	104	41.38	odd	8	inner
287.2.l.a.202.16	yes	104	287.202	even	8	inner