Embedded newform 287.2.bc.a.2.14

• Label: 287.2.bc.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0378756 - 0.178191i) q^{2} +(1.65415 + 0.443227i) q^{3} +(1.79677 + 0.799975i) q^{4} +(-1.19075 + 0.125152i) q^{5} +(0.141631 - 0.277966i) q^{6} +(2.15498 - 1.53494i) q^{7} +(0.424758 - 0.584629i) q^{8} +(-0.0583256 - 0.0336743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0378756	−	0.178191i	0.0267821	−	0.126000i	−0.962726	−	0.270480i	\(-0.912818\pi\)
							0.989508	+	0.144480i	\(0.0461509\pi\)
\(3\)	1.65415	+	0.443227i	0.955022	+	0.255897i	0.702492	−	0.711692i	\(-0.252071\pi\)
							0.252530	+	0.967589i	\(0.418737\pi\)
\(5\)	−1.19075	+	0.125152i	−0.532518	+	0.0559699i	−0.366972	−	0.930232i	\(-0.619605\pi\)
							−0.165546	+	0.986202i	\(0.552939\pi\)
\(7\)	2.15498	−	1.53494i	0.814507	−	0.580154i
							\(11\)	−0.581508	−	0.718103i	−0.175331	−	0.216516i	0.681903	−	0.731443i	\(-0.261153\pi\)
−0.857234	+	0.514926i	\(0.827819\pi\)
\(13\)	−0.732554	−	0.373255i	−0.203174	−	0.103522i	0.349443	−	0.936958i	\(-0.386371\pi\)
							−0.552617	+	0.833435i	\(0.686371\pi\)
\(17\)	−1.31136	+	1.06192i	−0.318052	+	0.257554i	−0.775030	−	0.631924i	\(-0.782265\pi\)
							0.456978	+	0.889478i	\(0.348932\pi\)
\(19\)	−0.240174	+	4.58279i	−0.0550996	+	1.05136i	0.821520	+	0.570180i	\(0.193126\pi\)
							−0.876620	+	0.481184i	\(0.840207\pi\)
\(23\)	−1.69256	−	0.359765i	−0.352924	−	0.0750162i	0.0280386	−	0.999607i	\(-0.491074\pi\)
							−0.380962	+	0.924591i	\(0.624407\pi\)
\(29\)	0.683234	−	4.31377i	0.126873	−	0.801047i	−0.839397	−	0.543518i	\(-0.817092\pi\)
							0.966271	−	0.257529i	\(-0.0829082\pi\)
\(31\)	−0.811837	+	7.72411i	−0.145810	+	1.38729i	0.639786	+	0.768553i	\(0.279023\pi\)
							−0.785597	+	0.618739i	\(0.787644\pi\)
\(37\)	−0.412030	−	3.92020i	−0.0677373	−	0.644477i	−0.974739	−	0.223349i	\(-0.928301\pi\)
							0.907001	−	0.421128i	\(-0.138366\pi\)
\(41\)	−3.67813	−	5.24131i	−0.574428	−	0.818555i
							\(43\)	8.58790	−	2.79038i	1.30964	−	0.425529i	0.430717	−	0.902487i	\(-0.358261\pi\)
0.878926	+	0.476958i	\(0.158261\pi\)
\(47\)	1.40313	+	2.16062i	0.204667	+	0.315159i	0.926067	−	0.377359i	\(-0.123168\pi\)
							−0.721400	+	0.692519i	\(0.756501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.14	✓	416
7.4	even	3	inner	287.2.bc.a.207.14	yes	416
41.21	even	20	inner	287.2.bc.a.226.14	yes	416
287.144	even	60	inner	287.2.bc.a.144.14	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.14	✓	416	1.1	even	1	trivial
287.2.bc.a.144.14	yes	416	287.144	even	60	inner
287.2.bc.a.207.14	yes	416	7.4	even	3	inner
287.2.bc.a.226.14	yes	416	41.21	even	20	inner