Embedded newform 287.2.bc.a.2.7

• Label: 287.2.bc.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.328452 + 1.54525i) q^{2} +(0.684405 + 0.183386i) q^{3} +(-0.452815 - 0.201606i) q^{4} +(-3.49854 + 0.367712i) q^{5} +(-0.508171 + 0.997341i) q^{6} +(-2.38756 - 1.13999i) q^{7} +(-1.39687 + 1.92263i) q^{8} +(-2.16330 - 1.24898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.328452	+	1.54525i	−0.232251	+	1.09265i	0.695231	+	0.718786i	\(0.255302\pi\)
							−0.927482	+	0.373868i	\(0.878031\pi\)
\(3\)	0.684405	+	0.183386i	0.395142	+	0.105878i	0.450918	−	0.892565i	\(-0.351097\pi\)
							−0.0557764	+	0.998443i	\(0.517763\pi\)
\(5\)	−3.49854	+	0.367712i	−1.56460	+	0.164446i	−0.846840	−	0.531848i	\(-0.821498\pi\)
							−0.717757	+	0.696294i	\(0.754831\pi\)
\(7\)	−2.38756	−	1.13999i	−0.902411	−	0.430876i
							\(11\)	−0.967296	−	1.19451i	−0.291651	−	0.360159i	0.610169	−	0.792271i	\(-0.291102\pi\)
−0.901820	+	0.432113i	\(0.857768\pi\)
\(13\)	2.70731	+	1.37944i	0.750872	+	0.382589i	0.787157	−	0.616753i	\(-0.211552\pi\)
							−0.0362844	+	0.999342i	\(0.511552\pi\)
\(17\)	−3.22831	+	2.61424i	−0.782981	+	0.634046i	−0.935127	−	0.354312i	\(-0.884715\pi\)
							0.152146	+	0.988358i	\(0.451382\pi\)
\(19\)	−0.352635	+	6.72868i	−0.0809001	+	1.54367i	0.593218	+	0.805042i	\(0.297857\pi\)
							−0.674118	+	0.738624i	\(0.735476\pi\)
\(23\)	4.73524	+	1.00651i	0.987367	+	0.209871i	0.673171	−	0.739487i	\(-0.264932\pi\)
							0.314196	+	0.949358i	\(0.398265\pi\)
\(29\)	−1.17860	+	7.44136i	−0.218860	+	1.38183i	0.596380	+	0.802702i	\(0.296605\pi\)
							−0.815240	+	0.579124i	\(0.803395\pi\)
\(31\)	0.137148	−	1.30488i	0.0246325	−	0.234363i	−0.975278	−	0.220982i	\(-0.929074\pi\)
							0.999910	−	0.0133810i	\(-0.00425943\pi\)
\(37\)	−0.744646	−	7.08483i	−0.122419	−	1.16474i	−0.867385	−	0.497637i	\(-0.834201\pi\)
							0.744966	−	0.667102i	\(-0.232466\pi\)
\(41\)	−4.99835	+	4.00207i	−0.780611	+	0.625018i
							\(43\)	4.90530	−	1.59383i	0.748051	−	0.243057i	0.0899089	−	0.995950i	\(-0.471342\pi\)
0.658143	+	0.752893i	\(0.271342\pi\)
\(47\)	−4.72882	−	7.28174i	−0.689769	−	1.06215i	−0.993982	−	0.109547i	\(-0.965060\pi\)
							0.304213	−	0.952604i	\(-0.401607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.7	✓	416
7.4	even	3	inner	287.2.bc.a.207.7	yes	416
41.21	even	20	inner	287.2.bc.a.226.7	yes	416
287.144	even	60	inner	287.2.bc.a.144.7	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.7	✓	416	1.1	even	1	trivial
287.2.bc.a.144.7	yes	416	287.144	even	60	inner
287.2.bc.a.207.7	yes	416	7.4	even	3	inner
287.2.bc.a.226.7	yes	416	41.21	even	20	inner