Embedded newform 287.2.s.a.16.14

• Label: 287.2.s.a.16.14
• 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			16.14
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0783279 + 0.0869920i) q^{2} +(1.70494 + 2.95304i) q^{3} +(0.207625 - 1.97542i) q^{4} +(0.329616 - 0.146754i) q^{5} +(-0.123347 + 0.379622i) q^{6} +(1.89833 - 1.84291i) q^{7} +(0.377514 - 0.274280i) q^{8} +(-4.31364 + 7.47144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0783279	+	0.0869920i	0.0553862	+	0.0615126i	0.770201	−	0.637801i	\(-0.220156\pi\)
							−0.714815	+	0.699314i	\(0.753489\pi\)
\(3\)	1.70494	+	2.95304i	0.984347	+	1.70494i	0.644801	+	0.764351i	\(0.276940\pi\)
							0.339547	+	0.940589i	\(0.389726\pi\)
\(5\)	0.329616	−	0.146754i	0.147409	−	0.0656305i	−0.331706	−	0.943383i	\(-0.607624\pi\)
							0.479115	+	0.877752i	\(0.340958\pi\)
\(7\)	1.89833	−	1.84291i	0.717503	−	0.696556i
							\(11\)	2.64516	+	1.17770i	0.797546	+	0.355091i	0.764714	−	0.644370i	\(-0.222880\pi\)
0.0328328	+	0.999461i	\(0.489547\pi\)
\(13\)	0.498751	−	1.53500i	0.138329	−	0.425732i	−0.857764	−	0.514043i	\(-0.828147\pi\)
							0.996093	+	0.0883114i	\(0.0281471\pi\)
\(17\)	−2.85697	−	1.27201i	−0.692918	−	0.308507i	0.0298926	−	0.999553i	\(-0.490483\pi\)
							−0.722811	+	0.691046i	\(0.757150\pi\)
\(19\)	−4.17917	−	0.888310i	−0.958767	−	0.203792i	−0.298153	−	0.954518i	\(-0.596371\pi\)
							−0.660614	+	0.750726i	\(0.729704\pi\)
\(23\)	1.45900	+	1.62039i	0.304223	+	0.337874i	0.875800	−	0.482675i	\(-0.160335\pi\)
							−0.571577	+	0.820549i	\(0.693668\pi\)
\(29\)	3.51769	+	2.55575i	0.653218	+	0.474591i	0.864366	−	0.502864i	\(-0.167720\pi\)
							−0.211148	+	0.977454i	\(0.567720\pi\)
\(31\)	−4.27044	−	1.90132i	−0.766993	−	0.341487i	−0.0143337	−	0.999897i	\(-0.504563\pi\)
							−0.752660	+	0.658410i	\(0.771229\pi\)
\(37\)	1.67498	−	0.745747i	0.275364	−	0.122600i	−0.264409	−	0.964411i	\(-0.585177\pi\)
							0.539773	+	0.841811i	\(0.318510\pi\)
\(41\)	1.94455	−	6.10072i	0.303687	−	0.952772i
							\(43\)	0.978380	−	3.01114i	0.149201	−	0.459195i	−0.848326	−	0.529475i	\(-0.822389\pi\)
0.997527	+	0.0702795i	\(0.0223891\pi\)
\(47\)	−6.27049	−	6.96409i	−0.914645	−	1.01582i	−0.999811	−	0.0194499i	\(-0.993809\pi\)
							0.0851654	−	0.996367i	\(-0.472858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.14	✓	208
7.4	even	3	inner	287.2.s.a.221.13	yes	208
41.18	even	5	inner	287.2.s.a.100.13	yes	208
287.18	even	15	inner	287.2.s.a.18.14	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.14	✓	208	1.1	even	1	trivial
287.2.s.a.18.14	yes	208	287.18	even	15	inner
287.2.s.a.100.13	yes	208	41.18	even	5	inner
287.2.s.a.221.13	yes	208	7.4	even	3	inner