Embedded newform 287.3.q.a.73.7

• Label: 287.3.q.a.73.7
• Level: $287$
• Weight: $3$
• Character: 287.73
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.7
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.81671 + 1.62623i) q^{2} +(1.49616 - 0.400895i) q^{3} +(3.28926 - 5.69716i) q^{4} +(4.01378 + 6.95206i) q^{5} +(-3.56231 + 3.56231i) q^{6} +(-0.176070 - 6.99779i) q^{7} +8.38651i q^{8} +(-5.71645 + 3.30039i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.81671	+	1.62623i	−1.40836	+	0.813116i	−0.995230	−	0.0975576i	\(-0.968897\pi\)
							−0.413128	+	0.910673i	\(0.635564\pi\)
\(3\)	1.49616	−	0.400895i	0.498720	−	0.133632i	−0.000685749	−	1.00000i	\(-0.500218\pi\)
							0.499406	+	0.866368i	\(0.333552\pi\)
\(5\)	4.01378	+	6.95206i	0.802755	+	1.39041i	0.917796	+	0.397052i	\(0.129967\pi\)
							−0.115041	+	0.993361i	\(0.536700\pi\)
\(7\)	−0.176070	−	6.99779i	−0.0251529	−	0.999684i
							\(11\)	5.40268	+	20.1631i	0.491153	+	1.83301i	0.550591	+	0.834775i	\(0.314403\pi\)
−0.0594377	+	0.998232i	\(0.518931\pi\)
\(13\)	−11.3345	−	11.3345i	−0.871882	−	0.871882i	0.120796	−	0.992677i	\(-0.461455\pi\)
							−0.992677	+	0.120796i	\(0.961455\pi\)
\(17\)	−1.51260	−	5.64509i	−0.0889763	−	0.332064i	0.907061	−	0.420999i	\(-0.138320\pi\)
							−0.996037	+	0.0889349i	\(0.971654\pi\)
\(19\)	21.9764	+	5.88855i	1.15665	+	0.309924i	0.785627	−	0.618700i	\(-0.212340\pi\)
							0.371023	+	0.928624i	\(0.379007\pi\)
\(23\)	12.6736	+	21.9514i	0.551027	+	0.954407i	0.998201	+	0.0599599i	\(0.0190973\pi\)
							−0.447174	+	0.894447i	\(0.647569\pi\)
\(29\)	−13.1158	+	13.1158i	−0.452269	+	0.452269i	−0.896107	−	0.443838i	\(-0.853617\pi\)
							0.443838	+	0.896107i	\(0.353617\pi\)
\(31\)	6.51810	+	3.76323i	0.210261	+	0.121394i	0.601433	−	0.798923i	\(-0.294597\pi\)
							−0.391172	+	0.920318i	\(0.627930\pi\)
\(37\)	16.6039	+	28.7588i	0.448754	+	0.777264i	0.998305	−	0.0581957i	\(-0.0185347\pi\)
							−0.549552	+	0.835460i	\(0.685201\pi\)
\(41\)	−40.7500	+	4.52114i	−0.993901	+	0.110272i
							\(43\)		−	19.0675i		−	0.443431i	−0.975111	−	0.221715i	\(-0.928834\pi\)
0.975111	−	0.221715i	\(-0.0711656\pi\)
\(47\)	−71.7265	−	19.2191i	−1.52610	−	0.408916i	−0.604353	−	0.796717i	\(-0.706568\pi\)
							−0.921743	+	0.387801i	\(0.873235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.7	✓	216
7.5	odd	6	inner	287.3.q.a.278.48	yes	216
41.9	even	4	inner	287.3.q.a.255.48	yes	216
287.173	odd	12	inner	287.3.q.a.173.7	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.7	✓	216	1.1	even	1	trivial
287.3.q.a.173.7	yes	216	287.173	odd	12	inner
287.3.q.a.255.48	yes	216	41.9	even	4	inner
287.3.q.a.278.48	yes	216	7.5	odd	6	inner