Embedded newform 287.2.z.a.4.2

• Label: 287.2.z.a.4.2
• Level: $287$
• Weight: $2$
• Character: 287.4
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.z (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33544 - 1.03981i) q^{2} +(2.55694 + 1.47625i) q^{3} +(3.03484 + 3.37053i) q^{4} +(-1.62101 + 0.344555i) q^{5} +(-4.43657 - 6.10641i) q^{6} +(0.110850 + 2.64343i) q^{7} +(-2.00302 - 6.16466i) q^{8} +(2.85861 + 4.95126i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33544	−	1.03981i	−1.65141	−	0.735255i	−0.651679	−	0.758495i	\(-0.725935\pi\)
							−0.999730	+	0.0232404i	\(0.992602\pi\)
\(3\)	2.55694	+	1.47625i	1.47625	+	0.852312i	0.999641	−	0.0268050i	\(-0.00853333\pi\)
							0.476606	+	0.879117i	\(0.341867\pi\)
\(5\)	−1.62101	+	0.344555i	−0.724936	+	0.154090i	−0.555577	−	0.831465i	\(-0.687503\pi\)
							−0.169358	+	0.985555i	\(0.554169\pi\)
\(7\)	0.110850	+	2.64343i	0.0418972	+	0.999122i
							\(11\)	0.933352	−	4.39107i	0.281416	−	1.32396i	−0.579400	−	0.815043i	\(-0.696713\pi\)
0.860816	−	0.508915i	\(-0.169953\pi\)
\(13\)	1.68746	+	2.32259i	0.468018	+	0.644171i	0.976147	−	0.217109i	\(-0.0696629\pi\)
							−0.508129	+	0.861281i	\(0.669663\pi\)
\(17\)	−1.34437	+	6.32477i	−0.326058	+	1.53398i	0.444011	+	0.896021i	\(0.353555\pi\)
							−0.770069	+	0.637961i	\(0.779778\pi\)
\(19\)	−5.11485	−	0.537592i	−1.17343	−	0.123332i	−0.502312	−	0.864686i	\(-0.667517\pi\)
							−0.671114	+	0.741354i	\(0.734184\pi\)
\(23\)	0.615143	+	0.273880i	0.128266	+	0.0571078i	0.469866	−	0.882738i	\(-0.344302\pi\)
							−0.341600	+	0.939845i	\(0.610969\pi\)
\(29\)	5.10211	+	1.65778i	0.947439	+	0.307841i	0.741674	−	0.670760i	\(-0.234032\pi\)
							0.205764	+	0.978602i	\(0.434032\pi\)
\(31\)	4.09652	+	0.870742i	0.735757	+	0.156390i	0.560525	−	0.828137i	\(-0.310599\pi\)
							0.175231	+	0.984527i	\(0.443933\pi\)
\(37\)	6.99810	−	1.48749i	1.15048	−	0.244542i	0.407082	−	0.913392i	\(-0.366546\pi\)
							0.743398	+	0.668850i	\(0.233213\pi\)
\(41\)	−6.15134	+	1.77792i	−0.960678	+	0.277665i
							\(43\)	2.31238	−	1.68004i	0.352634	−	0.256204i	−0.397339	−	0.917672i	\(-0.630066\pi\)
0.749973	+	0.661468i	\(0.230066\pi\)
\(47\)	1.62803	−	3.65662i	0.237473	−	0.533373i	−0.755016	−	0.655706i	\(-0.772371\pi\)
							0.992489	+	0.122333i	\(0.0390377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.2	✓	208
7.2	even	3	inner	287.2.z.a.86.25	yes	208
41.31	even	10	inner	287.2.z.a.277.25	yes	208
287.72	even	30	inner	287.2.z.a.72.2	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.2	✓	208	1.1	even	1	trivial
287.2.z.a.72.2	yes	208	287.72	even	30	inner
287.2.z.a.86.25	yes	208	7.2	even	3	inner
287.2.z.a.277.25	yes	208	41.31	even	10	inner