Embedded newform 287.2.bc.a.2.9

• Label: 287.2.bc.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.269531 + 1.26804i) q^{2} +(2.47947 + 0.664373i) q^{3} +(0.291806 + 0.129921i) q^{4} +(2.01290 - 0.211565i) q^{5} +(-1.51075 + 2.96501i) q^{6} +(-1.64461 - 2.07250i) q^{7} +(-1.76737 + 2.43258i) q^{8} +(3.10833 + 1.79459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.269531	+	1.26804i	−0.190587	+	0.896641i	0.774063	+	0.633108i	\(0.218221\pi\)
							−0.964650	+	0.263533i	\(0.915112\pi\)
\(3\)	2.47947	+	0.664373i	1.43153	+	0.383576i	0.889557	−	0.456823i	\(-0.151013\pi\)
							0.541968	+	0.840399i	\(0.317679\pi\)
\(5\)	2.01290	−	0.211565i	0.900198	−	0.0946146i	0.356888	−	0.934147i	\(-0.383838\pi\)
							0.543310	+	0.839532i	\(0.317171\pi\)
\(7\)	−1.64461	−	2.07250i	−0.621606	−	0.783330i
							\(11\)	−0.711135	−	0.878178i	−0.214415	−	0.264781i	0.658579	−	0.752512i	\(-0.271158\pi\)
−0.872994	+	0.487731i	\(0.837825\pi\)
\(13\)	−3.35890	−	1.71145i	−0.931592	−	0.474670i	−0.0787825	−	0.996892i	\(-0.525103\pi\)
							−0.852810	+	0.522222i	\(0.825103\pi\)
\(17\)	0.828552	−	0.670948i	0.200953	−	0.162729i	−0.523562	−	0.851988i	\(-0.675397\pi\)
							0.724515	+	0.689259i	\(0.242064\pi\)
\(19\)	0.141555	−	2.70103i	0.0324749	−	0.619658i	−0.933113	−	0.359583i	\(-0.882919\pi\)
							0.965588	−	0.260076i	\(-0.0837476\pi\)
\(23\)	−1.13554	−	0.241365i	−0.236775	−	0.0503282i	0.0879955	−	0.996121i	\(-0.471954\pi\)
							−0.324771	+	0.945793i	\(0.605287\pi\)
\(29\)	−0.481429	+	3.03962i	−0.0893991	+	0.564444i	0.901809	+	0.432134i	\(0.142239\pi\)
							−0.991208	+	0.132310i	\(0.957761\pi\)
\(31\)	−0.109639	+	1.04315i	−0.0196918	+	0.187355i	−0.999946	−	0.0103864i	\(-0.996694\pi\)
							0.980254	+	0.197741i	\(0.0633605\pi\)
\(37\)	1.09183	+	10.3881i	0.179496	+	1.70779i	0.599589	+	0.800308i	\(0.295331\pi\)
							−0.420093	+	0.907481i	\(0.638003\pi\)
\(41\)	−1.41340	−	6.24518i	−0.220735	−	0.975334i
							\(43\)	−0.518266	+	0.168395i	−0.0790348	+	0.0256800i	−0.348268	−	0.937395i	\(-0.613230\pi\)
0.269233	+	0.963075i	\(0.413230\pi\)
\(47\)	4.13694	+	6.37033i	0.603434	+	0.929207i	0.999938	+	0.0111741i	\(0.00355691\pi\)
							−0.396503	+	0.918033i	\(0.629776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.9	✓	416
7.4	even	3	inner	287.2.bc.a.207.9	yes	416
41.21	even	20	inner	287.2.bc.a.226.9	yes	416
287.144	even	60	inner	287.2.bc.a.144.9	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.9	✓	416	1.1	even	1	trivial
287.2.bc.a.144.9	yes	416	287.144	even	60	inner
287.2.bc.a.207.9	yes	416	7.4	even	3	inner
287.2.bc.a.226.9	yes	416	41.21	even	20	inner