Embedded newform 287.3.bd.a.5.20

• Label: 287.3.bd.a.5.20
• Level: $287$
• Weight: $3$
• Character: 287.5
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(54\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22238 + 0.128477i) q^{2} +(-2.41703 - 0.647641i) q^{3} +(-2.43489 + 0.517552i) q^{4} +(6.07573 + 6.74778i) q^{5} +(3.03773 + 0.481129i) q^{6} +(6.66096 + 2.15212i) q^{7} +(7.58568 - 2.46474i) q^{8} +(-2.37164 - 1.36927i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22238	+	0.128477i	−0.611189	+	0.0642385i	−0.405066	−	0.914288i	\(-0.632751\pi\)
							−0.206123	+	0.978526i	\(0.566085\pi\)
\(3\)	−2.41703	−	0.647641i	−0.805676	−	0.215880i	−0.167602	−	0.985855i	\(-0.553602\pi\)
							−0.638075	+	0.769975i	\(0.720269\pi\)
\(5\)	6.07573	+	6.74778i	1.21515	+	1.34956i	0.918923	+	0.394437i	\(0.129060\pi\)
							0.296222	+	0.955119i	\(0.404273\pi\)
\(7\)	6.66096	+	2.15212i	0.951566	+	0.307445i
							\(11\)	−9.18167	+	0.481191i	−0.834697	+	0.0437446i	−0.464895	−	0.885366i	\(-0.653908\pi\)
−0.369802	+	0.929111i	\(0.620575\pi\)
\(13\)	0.525377	−	3.31710i	0.0404136	−	0.255162i	−0.959207	−	0.282705i	\(-0.908768\pi\)
							0.999621	+	0.0275431i	\(0.00876837\pi\)
\(17\)	18.9903	−	0.995238i	1.11707	−	0.0585434i	0.515153	−	0.857098i	\(-0.327735\pi\)
							0.601922	+	0.798555i	\(0.294402\pi\)
\(19\)	2.91158	+	1.11765i	0.153241	+	0.0588237i	0.433780	−	0.901019i	\(-0.357180\pi\)
							−0.280538	+	0.959843i	\(0.590513\pi\)
\(23\)	4.03328	+	38.3741i	0.175360	+	1.66844i	0.629115	+	0.777313i	\(0.283418\pi\)
							−0.453754	+	0.891127i	\(0.649916\pi\)
\(29\)	−36.2875	+	18.4894i	−1.25129	+	0.637565i	−0.948887	−	0.315615i	\(-0.897789\pi\)
							−0.302404	+	0.953180i	\(0.597789\pi\)
\(31\)	−29.6789	−	26.7230i	−0.957384	−	0.862033i	0.0331286	−	0.999451i	\(-0.489453\pi\)
							−0.990513	+	0.137418i	\(0.956120\pi\)
\(37\)	21.8816	+	24.3020i	0.591394	+	0.656810i	0.962341	−	0.271844i	\(-0.0876336\pi\)
							−0.370947	+	0.928654i	\(0.620967\pi\)
\(41\)	−2.34367	+	40.9330i	−0.0571627	+	0.998365i
							\(43\)	−38.0133	−	52.3208i	−0.884030	−	1.21676i	−0.975288	−	0.220936i	\(-0.929089\pi\)
0.0912582	−	0.995827i	\(-0.470911\pi\)
\(47\)	−2.86395	+	3.53668i	−0.0609350	+	0.0752485i	−0.806702	−	0.590958i	\(-0.798750\pi\)
							0.745767	+	0.666207i	\(0.232083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.20	✓	864
7.3	odd	6	inner	287.3.bd.a.87.35	yes	864
41.33	even	20	inner	287.3.bd.a.33.35	yes	864
287.115	odd	60	inner	287.3.bd.a.115.20	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.20	✓	864	1.1	even	1	trivial
287.3.bd.a.33.35	yes	864	41.33	even	20	inner
287.3.bd.a.87.35	yes	864	7.3	odd	6	inner
287.3.bd.a.115.20	yes	864	287.115	odd	60	inner