Embedded newform 287.2.z.a.4.20

• Label: 287.2.z.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47672 + 0.657478i) q^{2} +(2.34477 + 1.35375i) q^{3} +(0.410161 + 0.455530i) q^{4} +(0.683534 - 0.145290i) q^{5} +(2.57250 + 3.54075i) q^{6} +(-1.72845 - 2.00311i) q^{7} +(-0.692842 - 2.13235i) q^{8} +(2.16529 + 3.75040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47672	+	0.657478i	1.04420	+	0.464907i	0.855866	−	0.517198i	\(-0.173025\pi\)
							0.188332	+	0.982105i	\(0.439692\pi\)
\(3\)	2.34477	+	1.35375i	1.35375	+	0.781590i	0.988773	−	0.149426i	\(-0.0477425\pi\)
							0.364980	+	0.931015i	\(0.381076\pi\)
\(5\)	0.683534	−	0.145290i	0.305686	−	0.0649755i	−0.0525150	−	0.998620i	\(-0.516724\pi\)
							0.358201	+	0.933645i	\(0.383390\pi\)
\(7\)	−1.72845	−	2.00311i	−0.653293	−	0.757105i
							\(11\)	−0.816400	+	3.84086i	−0.246154	+	1.15806i	0.665273	+	0.746600i	\(0.268315\pi\)
−0.911427	+	0.411463i	\(0.865018\pi\)
\(13\)	−2.25165	−	3.09913i	−0.624495	−	0.859544i	0.373175	−	0.927761i	\(-0.378269\pi\)
							−0.997671	+	0.0682170i	\(0.978269\pi\)
\(17\)	−0.768064	+	3.61346i	−0.186283	+	0.876392i	0.781363	+	0.624077i	\(0.214525\pi\)
							−0.967645	+	0.252314i	\(0.918808\pi\)
\(19\)	−3.66022	−	0.384704i	−0.839711	−	0.0882572i	−0.325099	−	0.945680i	\(-0.605398\pi\)
							−0.514612	+	0.857423i	\(0.672064\pi\)
\(23\)	3.11672	+	1.38765i	0.649880	+	0.289345i	0.705078	−	0.709130i	\(-0.250912\pi\)
							−0.0551975	+	0.998475i	\(0.517579\pi\)
\(29\)	−3.66803	−	1.19181i	−0.681136	−	0.221314i	−0.0520432	−	0.998645i	\(-0.516573\pi\)
							−0.629093	+	0.777330i	\(0.716573\pi\)
\(31\)	1.58062	+	0.335972i	0.283889	+	0.0603424i	0.347655	−	0.937622i	\(-0.386978\pi\)
							−0.0637666	+	0.997965i	\(0.520311\pi\)
\(37\)	7.41517	−	1.57614i	1.21905	−	0.259116i	0.446913	−	0.894577i	\(-0.352523\pi\)
							0.772133	+	0.635461i	\(0.219190\pi\)
\(41\)	5.36013	+	3.50271i	0.837112	+	0.547032i
							\(43\)	8.76270	−	6.36647i	1.33630	−	0.970878i	0.336728	−	0.941602i	\(-0.390680\pi\)
0.999571	−	0.0292759i	\(-0.00932016\pi\)
\(47\)	−1.57286	+	3.53271i	−0.229426	+	0.515299i	−0.991173	−	0.132576i	\(-0.957675\pi\)
							0.761747	+	0.647875i	\(0.224342\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.20	✓	208
7.2	even	3	inner	287.2.z.a.86.7	yes	208
41.31	even	10	inner	287.2.z.a.277.7	yes	208
287.72	even	30	inner	287.2.z.a.72.20	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.20	✓	208	1.1	even	1	trivial
287.2.z.a.72.20	yes	208	287.72	even	30	inner
287.2.z.a.86.7	yes	208	7.2	even	3	inner
287.2.z.a.277.7	yes	208	41.31	even	10	inner