Embedded newform 287.2.w.a.3.18

• Label: 287.2.w.a.3.18
• Level: $287$
• Weight: $2$
• Character: 287.3
• 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.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.18
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.792178 + 0.212263i) q^{2} +(1.39160 - 1.06781i) q^{3} +(-1.14956 - 0.663699i) q^{4} +(-1.93992 - 0.519800i) q^{5} +(1.32905 - 0.550510i) q^{6} +(-1.09995 - 2.40626i) q^{7} +(-1.92961 - 1.92961i) q^{8} +(0.0198671 - 0.0741452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.792178	+	0.212263i	0.560154	+	0.150093i	0.527777	−	0.849383i	\(-0.323026\pi\)
							0.0323771	+	0.999476i	\(0.489692\pi\)
\(3\)	1.39160	−	1.06781i	0.803439	−	0.616500i	−0.123230	−	0.992378i	\(-0.539325\pi\)
							0.926669	+	0.375878i	\(0.122659\pi\)
\(5\)	−1.93992	−	0.519800i	−0.867559	−	0.232462i	−0.202527	−	0.979277i	\(-0.564915\pi\)
							−0.665032	+	0.746815i	\(0.731582\pi\)
\(7\)	−1.09995	−	2.40626i	−0.415743	−	0.909482i
							\(11\)	4.74543	−	3.64130i	1.43080	−	1.09789i	0.452250	−	0.891891i	\(-0.350622\pi\)
0.978552	−	0.206001i	\(-0.0660452\pi\)
\(13\)	0.723591	+	0.299721i	0.200688	+	0.0831277i	0.480764	−	0.876850i	\(-0.340360\pi\)
							−0.280076	+	0.959978i	\(0.590360\pi\)
\(17\)	0.180193	−	1.36870i	0.0437032	−	0.331959i	−0.955601	−	0.294664i	\(-0.904792\pi\)
							0.999304	−	0.0372955i	\(-0.0118743\pi\)
\(19\)	4.35949	+	3.34516i	1.00014	+	0.767432i	0.972670	−	0.232191i	\(-0.0745894\pi\)
							0.0274664	+	0.999623i	\(0.491256\pi\)
\(23\)	−0.516101	+	0.297971i	−0.107614	+	0.0621312i	−0.552841	−	0.833287i	\(-0.686456\pi\)
							0.445227	+	0.895418i	\(0.353123\pi\)
\(29\)	−2.29982	+	5.55226i	−0.427066	+	1.03103i	0.553147	+	0.833083i	\(0.313427\pi\)
							−0.980213	+	0.197945i	\(0.936573\pi\)
\(31\)	−0.810661	+	1.40411i	−0.145599	+	0.252185i	−0.929596	−	0.368579i	\(-0.879844\pi\)
							0.783997	+	0.620764i	\(0.213178\pi\)
\(37\)	−3.71047	−	6.42672i	−0.609997	−	1.05655i	−0.991240	−	0.132071i	\(-0.957837\pi\)
							0.381243	−	0.924475i	\(-0.375496\pi\)
\(41\)	4.87558	+	4.15074i	0.761438	+	0.648237i
							\(43\)	7.78446	−	7.78446i	1.18712	−	1.18712i	0.209259	−	0.977860i	\(-0.432895\pi\)
0.977860	−	0.209259i	\(-0.0671053\pi\)
\(47\)	0.634859	+	0.487144i	0.0926037	+	0.0710573i	0.654026	−	0.756472i	\(-0.273079\pi\)
							−0.561422	+	0.827530i	\(0.689746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.18	✓	208
7.5	odd	6	inner	287.2.w.a.208.9	yes	208
41.14	odd	8	inner	287.2.w.a.178.9	yes	208
287.96	even	24	inner	287.2.w.a.96.18	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.18	✓	208	1.1	even	1	trivial
287.2.w.a.96.18	yes	208	287.96	even	24	inner
287.2.w.a.178.9	yes	208	41.14	odd	8	inner
287.2.w.a.208.9	yes	208	7.5	odd	6	inner