Embedded newform 287.2.bc.a.2.20

• Label: 287.2.bc.a.2.20
• 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			2.20
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.340125 - 1.60016i) q^{2} +(1.76347 + 0.472522i) q^{3} +(-0.617737 - 0.275034i) q^{4} +(-0.439959 + 0.0462415i) q^{5} +(1.35591 - 2.66113i) q^{6} +(1.44808 + 2.21428i) q^{7} +(1.27292 - 1.75202i) q^{8} +(0.288490 + 0.166560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.340125	−	1.60016i	0.240504	−	1.13148i	−0.677688	−	0.735350i	\(-0.737018\pi\)
							0.918192	−	0.396135i	\(-0.129649\pi\)
\(3\)	1.76347	+	0.472522i	1.01814	+	0.272810i	0.729028	−	0.684484i	\(-0.239972\pi\)
							0.289115	+	0.957294i	\(0.406639\pi\)
\(5\)	−0.439959	+	0.0462415i	−0.196756	+	0.0206798i	−0.202394	−	0.979304i	\(-0.564872\pi\)
							0.00563799	+	0.999984i	\(0.498205\pi\)
\(7\)	1.44808	+	2.21428i	0.547324	+	0.836921i
							\(11\)	2.10573	+	2.60036i	0.634900	+	0.784037i	0.988477	−	0.151369i	\(-0.0483681\pi\)
−0.353577	+	0.935405i	\(0.615035\pi\)
\(13\)	−3.69337	−	1.88187i	−1.02436	−	0.521936i	−0.140691	−	0.990053i	\(-0.544933\pi\)
							−0.883666	+	0.468117i	\(0.844933\pi\)
\(17\)	−0.714452	+	0.578552i	−0.173280	+	0.140319i	−0.712037	−	0.702142i	\(-0.752227\pi\)
							0.538757	+	0.842461i	\(0.318894\pi\)
\(19\)	0.194852	−	3.71800i	0.0447022	−	0.852968i	−0.880629	−	0.473807i	\(-0.842879\pi\)
							0.925331	−	0.379161i	\(-0.123787\pi\)
\(23\)	1.58796	+	0.337531i	0.331112	+	0.0703801i	0.370468	−	0.928845i	\(-0.379197\pi\)
							−0.0393558	+	0.999225i	\(0.512531\pi\)
\(29\)	−1.63546	+	10.3259i	−0.303698	+	1.91747i	0.0855410	+	0.996335i	\(0.472738\pi\)
							−0.389239	+	0.921137i	\(0.627262\pi\)
\(31\)	0.752216	−	7.15686i	0.135102	−	1.28541i	−0.691398	−	0.722474i	\(-0.743005\pi\)
							0.826500	−	0.562936i	\(-0.190328\pi\)
\(37\)	0.0218647	+	0.208029i	0.00359453	+	0.0341997i	0.996172	−	0.0874132i	\(-0.0278600\pi\)
							−0.992578	+	0.121613i	\(0.961193\pi\)
\(41\)	−5.68318	−	2.94983i	−0.887563	−	0.460686i
							\(43\)	−1.14134	+	0.370844i	−0.174053	+	0.0565532i	−0.394747	−	0.918790i	\(-0.629168\pi\)
0.220694	+	0.975343i	\(0.429168\pi\)
\(47\)	1.80550	+	2.78022i	0.263359	+	0.405537i	0.945469	−	0.325712i	\(-0.105604\pi\)
							−0.682110	+	0.731250i	\(0.738937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.20	✓	416
7.4	even	3	inner	287.2.bc.a.207.20	yes	416
41.21	even	20	inner	287.2.bc.a.226.20	yes	416
287.144	even	60	inner	287.2.bc.a.144.20	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.20	✓	416	1.1	even	1	trivial
287.2.bc.a.144.20	yes	416	287.144	even	60	inner
287.2.bc.a.207.20	yes	416	7.4	even	3	inner
287.2.bc.a.226.20	yes	416	41.21	even	20	inner