Embedded newform 287.2.s.a.16.3

• Label: 287.2.s.a.16.3
• Level: $287$
• Weight: $2$
• Character: 287.16
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.3
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45256 - 1.61323i) q^{2} +(1.19642 + 2.07226i) q^{3} +(-0.283525 + 2.69756i) q^{4} +(0.0523239 - 0.0232961i) q^{5} +(1.60516 - 4.94017i) q^{6} +(2.62771 + 0.308433i) q^{7} +(1.25116 - 0.909021i) q^{8} +(-1.36284 + 2.36051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45256	−	1.61323i	−1.02711	−	1.14072i	−0.989951	−	0.141414i	\(-0.954835\pi\)
							−0.0371612	−	0.999309i	\(-0.511831\pi\)
\(3\)	1.19642	+	2.07226i	0.690753	+	1.19642i	0.971591	+	0.236664i	\(0.0760542\pi\)
							−0.280838	+	0.959755i	\(0.590613\pi\)
\(5\)	0.0523239	−	0.0232961i	0.0234000	−	0.0104183i	−0.395003	−	0.918680i	\(-0.629257\pi\)
							0.418403	+	0.908261i	\(0.362590\pi\)
\(7\)	2.62771	+	0.308433i	0.993182	+	0.116577i
							\(11\)	−2.06496	−	0.919377i	−0.622607	−	0.277203i	0.0710892	−	0.997470i	\(-0.477352\pi\)
−0.693697	+	0.720267i	\(0.744019\pi\)
\(13\)	−1.31319	+	4.04158i	−0.364214	+	1.12093i	0.586259	+	0.810124i	\(0.300600\pi\)
							−0.950472	+	0.310810i	\(0.899400\pi\)
\(17\)	6.57256	+	2.92629i	1.59408	+	0.709730i	0.995800	−	0.0915579i	\(-0.0291846\pi\)
							0.598279	+	0.801288i	\(0.295851\pi\)
\(19\)	−0.814704	−	0.173171i	−0.186906	−	0.0397281i	0.113508	−	0.993537i	\(-0.463791\pi\)
							−0.300414	+	0.953809i	\(0.597125\pi\)
\(23\)	0.681083	+	0.756419i	0.142016	+	0.157724i	0.809956	−	0.586490i	\(-0.199491\pi\)
							−0.667941	+	0.744214i	\(0.732824\pi\)
\(29\)	−3.80733	−	2.76619i	−0.707004	−	0.513668i	0.175202	−	0.984533i	\(-0.443942\pi\)
							−0.882206	+	0.470864i	\(0.843942\pi\)
\(31\)	6.73498	+	2.99861i	1.20964	+	0.538566i	0.909649	−	0.415377i	\(-0.136350\pi\)
							0.299989	+	0.953943i	\(0.403017\pi\)
\(37\)	2.92502	−	1.30230i	0.480871	−	0.214097i	−0.151961	−	0.988386i	\(-0.548559\pi\)
							0.632832	+	0.774289i	\(0.281892\pi\)
\(41\)	0.0973136	−	6.40238i	0.0151978	−	0.999885i
							\(43\)	0.214026	−	0.658703i	0.0326386	−	0.100451i	−0.933410	−	0.358811i	\(-0.883182\pi\)
0.966049	+	0.258360i	\(0.0831821\pi\)
\(47\)	−2.89311	−	3.21312i	−0.422003	−	0.468682i	0.494228	−	0.869332i	\(-0.335451\pi\)
							−0.916231	+	0.400650i	\(0.868784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.3	✓	208
7.4	even	3	inner	287.2.s.a.221.24	yes	208
41.18	even	5	inner	287.2.s.a.100.24	yes	208
287.18	even	15	inner	287.2.s.a.18.3	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.3	✓	208	1.1	even	1	trivial
287.2.s.a.18.3	yes	208	287.18	even	15	inner
287.2.s.a.100.24	yes	208	41.18	even	5	inner
287.2.s.a.221.24	yes	208	7.4	even	3	inner