Embedded newform 287.2.bc.a.2.6

• Label: 287.2.bc.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.413424 + 1.94501i) q^{2} +(0.0866205 + 0.0232099i) q^{3} +(-1.78504 - 0.794752i) q^{4} +(2.38115 - 0.250269i) q^{5} +(-0.0809544 + 0.158882i) q^{6} +(1.97323 - 1.76249i) q^{7} +(-0.0537962 + 0.0740442i) q^{8} +(-2.59111 - 1.49598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.413424	+	1.94501i	−0.292335	+	1.37533i	0.549455	+	0.835523i	\(0.314835\pi\)
							−0.841790	+	0.539805i	\(0.818498\pi\)
\(3\)	0.0866205	+	0.0232099i	0.0500104	+	0.0134002i	0.283737	−	0.958902i	\(-0.408426\pi\)
							−0.233727	+	0.972302i	\(0.575092\pi\)
\(5\)	2.38115	−	0.250269i	1.06488	−	0.111924i	0.444156	−	0.895950i	\(-0.353504\pi\)
							0.620728	+	0.784026i	\(0.286837\pi\)
\(7\)	1.97323	−	1.76249i	0.745811	−	0.666157i
							\(11\)	3.72089	+	4.59491i	1.12189	+	1.38542i	0.913237	+	0.407428i	\(0.133574\pi\)
0.208652	+	0.977990i	\(0.433092\pi\)
\(13\)	3.90850	+	1.99148i	1.08402	+	0.552337i	0.902342	−	0.431020i	\(-0.141846\pi\)
							0.181681	+	0.983358i	\(0.441846\pi\)
\(17\)	−1.35335	+	1.09592i	−0.328236	+	0.265800i	−0.779245	−	0.626719i	\(-0.784397\pi\)
							0.451010	+	0.892519i	\(0.351064\pi\)
\(19\)	0.147461	−	2.81372i	0.0338298	−	0.645511i	−0.928119	−	0.372283i	\(-0.878575\pi\)
							0.961949	−	0.273228i	\(-0.0880915\pi\)
\(23\)	−4.57288	−	0.971996i	−0.953512	−	0.202675i	−0.295213	−	0.955432i	\(-0.595390\pi\)
							−0.658299	+	0.752756i	\(0.728724\pi\)
\(29\)	−0.217509	+	1.37330i	−0.0403905	+	0.255015i	−0.999619	−	0.0276177i	\(-0.991208\pi\)
							0.959228	+	0.282633i	\(0.0912079\pi\)
\(31\)	−0.276929	+	2.63480i	−0.0497379	+	0.473224i	0.941096	+	0.338139i	\(0.109797\pi\)
							−0.990834	+	0.135085i	\(0.956869\pi\)
\(37\)	−0.616013	−	5.86098i	−0.101272	−	0.963538i	−0.920677	−	0.390324i	\(-0.872363\pi\)
							0.819406	−	0.573214i	\(-0.194304\pi\)
\(41\)	−5.31118	+	3.57651i	−0.829466	+	0.558557i
							\(43\)	−0.530321	+	0.172312i	−0.0808732	+	0.0262773i	−0.349174	−	0.937058i	\(-0.613538\pi\)
0.268301	+	0.963335i	\(0.413538\pi\)
\(47\)	−0.475651	−	0.732438i	−0.0693808	−	0.106837i	0.802277	−	0.596952i	\(-0.203622\pi\)
							−0.871658	+	0.490115i	\(0.836955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.6	✓	416
7.4	even	3	inner	287.2.bc.a.207.6	yes	416
41.21	even	20	inner	287.2.bc.a.226.6	yes	416
287.144	even	60	inner	287.2.bc.a.144.6	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.6	✓	416	1.1	even	1	trivial
287.2.bc.a.144.6	yes	416	287.144	even	60	inner
287.2.bc.a.207.6	yes	416	7.4	even	3	inner
287.2.bc.a.226.6	yes	416	41.21	even	20	inner