Embedded newform 287.2.bc.a.2.18

• Label: 287.2.bc.a.2.18
• 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			2.18
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.188832 - 0.888384i) q^{2} +(-3.02660 - 0.810975i) q^{3} +(1.07352 + 0.477963i) q^{4} +(3.12880 - 0.328850i) q^{5} +(-1.29198 + 2.53564i) q^{6} +(-1.47874 + 2.19393i) q^{7} +(1.69502 - 2.33299i) q^{8} +(5.90455 + 3.40899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.188832	−	0.888384i	0.133524	−	0.628182i	−0.859585	−	0.510992i	\(-0.829278\pi\)
							0.993110	−	0.117190i	\(-0.0373886\pi\)
\(3\)	−3.02660	−	0.810975i	−1.74741	−	0.468217i	−0.763338	−	0.645999i	\(-0.776441\pi\)
							−0.984070	+	0.177783i	\(0.943108\pi\)
\(5\)	3.12880	−	0.328850i	1.39924	−	0.147066i	0.625330	−	0.780361i	\(-0.284964\pi\)
							0.773911	+	0.633295i	\(0.218298\pi\)
\(7\)	−1.47874	+	2.19393i	−0.558910	+	0.829228i
							\(11\)	1.99847	+	2.46791i	0.602563	+	0.744103i	0.983582	−	0.180464i	\(-0.0577600\pi\)
−0.381019	+	0.924567i	\(0.624427\pi\)
\(13\)	−1.06979	−	0.545087i	−0.296708	−	0.151180i	0.299300	−	0.954159i	\(-0.403247\pi\)
							−0.596007	+	0.802979i	\(0.703247\pi\)
\(17\)	4.58603	−	3.71369i	1.11227	−	0.900702i	0.116560	−	0.993184i	\(-0.462813\pi\)
							0.995714	+	0.0924815i	\(0.0294799\pi\)
\(19\)	0.352622	−	6.72844i	0.0808971	−	1.54361i	−0.593254	−	0.805016i	\(-0.702157\pi\)
							0.674151	−	0.738594i	\(-0.264510\pi\)
\(23\)	−2.36728	−	0.503180i	−0.493611	−	0.104920i	−0.0456210	−	0.998959i	\(-0.514527\pi\)
							−0.447990	+	0.894039i	\(0.647860\pi\)
\(29\)	−0.253955	+	1.60341i	−0.0471584	+	0.297746i	−0.999983	−	0.00577417i	\(-0.998162\pi\)
							0.952825	+	0.303520i	\(0.0981620\pi\)
\(31\)	−0.632371	+	6.01661i	−0.113577	+	1.08061i	0.778162	+	0.628064i	\(0.216152\pi\)
							−0.891739	+	0.452551i	\(0.850514\pi\)
\(37\)	0.0342035	+	0.325425i	0.00562302	+	0.0534995i	0.996974	−	0.0777409i	\(-0.0247707\pi\)
							−0.991351	+	0.131240i	\(0.958104\pi\)
\(41\)	−4.52553	+	4.52986i	−0.706769	+	0.707445i
							\(43\)	9.84360	−	3.19838i	1.50114	−	0.487748i	0.560786	−	0.827961i	\(-0.310499\pi\)
0.940349	+	0.340212i	\(0.110499\pi\)
\(47\)	1.96488	+	3.02565i	0.286607	+	0.441336i	0.952423	−	0.304780i	\(-0.0985829\pi\)
							−0.665816	+	0.746116i	\(0.731916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.18	✓	416
7.4	even	3	inner	287.2.bc.a.207.18	yes	416
41.21	even	20	inner	287.2.bc.a.226.18	yes	416
287.144	even	60	inner	287.2.bc.a.144.18	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.18	✓	416	1.1	even	1	trivial
287.2.bc.a.144.18	yes	416	287.144	even	60	inner
287.2.bc.a.207.18	yes	416	7.4	even	3	inner
287.2.bc.a.226.18	yes	416	41.21	even	20	inner