Embedded newform 287.2.w.a.3.17

• Label: 287.2.w.a.3.17
• Level: $287$
• Weight: $2$
• Character: 287.3
• 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.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.772348 + 0.206950i) q^{2} +(2.10066 - 1.61189i) q^{3} +(-1.17836 - 0.680325i) q^{4} +(0.633392 + 0.169717i) q^{5} +(1.95602 - 0.810211i) q^{6} +(2.32510 + 1.26249i) q^{7} +(-1.90010 - 1.90010i) q^{8} +(1.03812 - 3.87430i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.772348	+	0.206950i	0.546133	+	0.146336i	0.521328	−	0.853357i	\(-0.325437\pi\)
							0.0248048	+	0.999692i	\(0.492104\pi\)
\(3\)	2.10066	−	1.61189i	1.21282	−	0.930627i	0.213886	−	0.976859i	\(-0.431388\pi\)
							0.998931	+	0.0462319i	\(0.0147213\pi\)
\(5\)	0.633392	+	0.169717i	0.283261	+	0.0758996i	0.397652	−	0.917536i	\(-0.369825\pi\)
							−0.114391	+	0.993436i	\(0.536492\pi\)
\(7\)	2.32510	+	1.26249i	0.878807	+	0.477178i
							\(11\)	−0.920613	+	0.706411i	−0.277575	+	0.212991i	−0.738162	−	0.674624i	\(-0.764306\pi\)
0.460586	+	0.887615i	\(0.347639\pi\)
\(13\)	−2.39996	−	0.994098i	−0.665630	−	0.275713i	0.0241753	−	0.999708i	\(-0.492304\pi\)
							−0.689806	+	0.723995i	\(0.742304\pi\)
\(17\)	−0.531213	+	4.03496i	−0.128838	+	0.978622i	0.797867	+	0.602833i	\(0.205962\pi\)
							−0.926705	+	0.375789i	\(0.877372\pi\)
\(19\)	0.119374	+	0.0915986i	0.0273862	+	0.0210142i	0.622364	−	0.782728i	\(-0.286172\pi\)
							−0.594978	+	0.803742i	\(0.702839\pi\)
\(23\)	4.86926	−	2.81127i	1.01531	−	0.586190i	0.102569	−	0.994726i	\(-0.467294\pi\)
							0.912742	+	0.408536i	\(0.133960\pi\)
\(29\)	−1.23926	+	2.99183i	−0.230124	+	0.555569i	−0.996192	−	0.0871914i	\(-0.972211\pi\)
							0.766067	+	0.642760i	\(0.222211\pi\)
\(31\)	−0.930952	+	1.61246i	−0.167204	+	0.289606i	−0.937436	−	0.348158i	\(-0.886807\pi\)
							0.770232	+	0.637764i	\(0.220140\pi\)
\(37\)	2.97551	+	5.15373i	0.489171	+	0.847269i	0.999922	−	0.0124596i	\(-0.00396613\pi\)
							−0.510752	+	0.859728i	\(0.670633\pi\)
\(41\)	−5.30060	−	3.59216i	−0.827815	−	0.561002i
							\(43\)	0.785872	−	0.785872i	0.119844	−	0.119844i	−0.644641	−	0.764485i	\(-0.722993\pi\)
0.764485	+	0.644641i	\(0.222993\pi\)
\(47\)	2.58957	+	1.98705i	0.377728	+	0.289841i	0.780124	−	0.625625i	\(-0.215156\pi\)
							−0.402396	+	0.915466i	\(0.631823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.17	✓	208
7.5	odd	6	inner	287.2.w.a.208.10	yes	208
41.14	odd	8	inner	287.2.w.a.178.10	yes	208
287.96	even	24	inner	287.2.w.a.96.17	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.17	✓	208	1.1	even	1	trivial
287.2.w.a.96.17	yes	208	287.96	even	24	inner
287.2.w.a.178.10	yes	208	41.14	odd	8	inner
287.2.w.a.208.10	yes	208	7.5	odd	6	inner