Embedded newform 287.2.l.a.27.15

• Label: 287.2.l.a.27.15
• 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			27.15
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.279520 + 0.279520i) q^{2} +(-0.739386 - 1.78504i) q^{3} -1.84374i q^{4} +(2.85060 + 2.85060i) q^{5} +(0.292280 - 0.705627i) q^{6} +(1.56158 + 2.13576i) q^{7} +(1.07440 - 1.07440i) q^{8} +(-0.518341 + 0.518341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

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	\(-0.936664\pi\)
							0.553385	−	0.832926i	\(-0.313336\pi\)
\(5\)	2.85060	+	2.85060i	1.27483	+	1.27483i	0.943524	+	0.331305i	\(0.107489\pi\)
							0.331305	+	0.943524i	\(0.392511\pi\)
\(7\)	1.56158	+	2.13576i	0.590220	+	0.807242i
							\(11\)	−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	0.833235	−	0.552919i	\(-0.186486\pi\)
							−0.980159	+	0.198213i	\(0.936486\pi\)
\(17\)	−1.48320	+	3.58077i	−0.359729	+	0.868463i	0.635608	+	0.772012i	\(0.280749\pi\)
							−0.995338	+	0.0964516i	\(0.969251\pi\)
\(19\)	2.65648	−	6.41330i	0.609438	−	1.47131i	−0.254176	−	0.967158i	\(-0.581804\pi\)
							0.863614	−	0.504154i	\(-0.168196\pi\)
\(23\)			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	\(-0.726103\pi\)
							−0.652079	+	0.758151i	\(0.726103\pi\)
\(37\)	−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	−0.999947	−	0.0103206i	\(-0.996715\pi\)
							0.714367	+	0.699771i	\(0.246715\pi\)

(See \(a_n\) instead)

Twists

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

    

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