Embedded newform 287.2.bc.a.2.12

• Label: 287.2.bc.a.2.12
• Level: $287$
• Weight: $2$
• Character: 287.2
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.bc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(26\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0934584 + 0.439687i) q^{2} +(-1.72460 - 0.462106i) q^{3} +(1.64250 + 0.731288i) q^{4} +(-2.14313 + 0.225252i) q^{5} +(0.364361 - 0.715098i) q^{6} +(-1.08772 - 2.41182i) q^{7} +(-1.00347 + 1.38116i) q^{8} +(0.162634 + 0.0938970i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0934584	+	0.439687i	−0.0660851	+	0.310906i	−0.998758	−	0.0498163i	\(-0.984136\pi\)
							0.932673	+	0.360722i	\(0.117470\pi\)
\(3\)	−1.72460	−	0.462106i	−0.995699	−	0.266797i	−0.276057	−	0.961141i	\(-0.589028\pi\)
							−0.719643	+	0.694344i	\(0.755694\pi\)
\(5\)	−2.14313	+	0.225252i	−0.958438	+	0.100736i	−0.570812	−	0.821081i	\(-0.693371\pi\)
							−0.387626	+	0.921817i	\(0.626705\pi\)
\(7\)	−1.08772	−	2.41182i	−0.411120	−	0.911581i
							\(11\)	0.737879	+	0.911205i	0.222479	+	0.274739i	0.876148	−	0.482042i	\(-0.160105\pi\)
−0.653669	+	0.756780i	\(0.726771\pi\)
\(13\)	−5.82296	−	2.96695i	−1.61500	−	0.822883i	−0.999393	−	0.0348282i	\(-0.988912\pi\)
							−0.615605	−	0.788055i	\(-0.711088\pi\)
\(17\)	0.0759395	−	0.0614946i	0.0184180	−	0.0149146i	−0.620067	−	0.784549i	\(-0.712895\pi\)
							0.638485	+	0.769634i	\(0.279561\pi\)
\(19\)	0.340538	−	6.49785i	0.0781248	−	1.49071i	−0.625548	−	0.780185i	\(-0.715125\pi\)
							0.703673	−	0.710524i	\(-0.251542\pi\)
\(23\)	−7.06174	−	1.50102i	−1.47248	−	0.312984i	−0.599353	−	0.800485i	\(-0.704575\pi\)
							−0.873122	+	0.487501i	\(0.837909\pi\)
\(29\)	0.447697	−	2.82665i	0.0831352	−	0.524895i	−0.910614	−	0.413258i	\(-0.864391\pi\)
							0.993749	−	0.111637i	\(-0.0356093\pi\)
\(31\)	−0.270961	+	2.57802i	−0.0486660	+	0.463026i	0.942867	+	0.333170i	\(0.108118\pi\)
							−0.991533	+	0.129856i	\(0.958548\pi\)
\(37\)	−0.799224	−	7.60411i	−0.131392	−	1.25011i	−0.839247	−	0.543751i	\(-0.817004\pi\)
							0.707855	−	0.706358i	\(-0.249663\pi\)
\(41\)	−0.549791	+	6.37948i	−0.0858630	+	0.996307i
							\(43\)	−3.90241	+	1.26797i	−0.595111	+	0.193363i	−0.591059	−	0.806628i	\(-0.701290\pi\)
−0.00405238	+	0.999992i	\(0.501290\pi\)
\(47\)	5.19305	+	7.99659i	0.757484	+	1.16642i	0.981430	+	0.191821i	\(0.0614393\pi\)
							−0.223946	+	0.974601i	\(0.571894\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.12	✓	416
7.4	even	3	inner	287.2.bc.a.207.12	yes	416
41.21	even	20	inner	287.2.bc.a.226.12	yes	416
287.144	even	60	inner	287.2.bc.a.144.12	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.12	✓	416	1.1	even	1	trivial
287.2.bc.a.144.12	yes	416	287.144	even	60	inner
287.2.bc.a.207.12	yes	416	7.4	even	3	inner
287.2.bc.a.226.12	yes	416	41.21	even	20	inner