Embedded newform 287.2.bc.a.2.11

• Label: 287.2.bc.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.163542 + 0.769404i) q^{2} +(-2.22695 - 0.596709i) q^{3} +(1.26185 + 0.561814i) q^{4} +(2.23699 - 0.235117i) q^{5} +(0.823309 - 1.61584i) q^{6} +(0.257315 - 2.63321i) q^{7} +(-1.56332 + 2.15173i) q^{8} +(2.00516 + 1.15768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.163542	+	0.769404i	−0.115642	+	0.544051i	0.881740	+	0.471735i	\(0.156372\pi\)
							−0.997382	+	0.0723154i	\(0.976961\pi\)
\(3\)	−2.22695	−	0.596709i	−1.28573	−	0.344510i	−0.449692	−	0.893184i	\(-0.648466\pi\)
							−0.836037	+	0.548673i	\(0.815133\pi\)
\(5\)	2.23699	−	0.235117i	1.00041	−	0.105147i	0.409860	−	0.912148i	\(-0.365578\pi\)
							0.590550	+	0.807001i	\(0.298911\pi\)
\(7\)	0.257315	−	2.63321i	0.0972560	−	0.995259i
							\(11\)	−3.44346	−	4.25232i	−1.03824	−	1.28212i	−0.958435	−	0.285313i	\(-0.907903\pi\)
−0.0798081	−	0.996810i	\(-0.525431\pi\)
\(13\)	5.66428	+	2.88610i	1.57099	+	0.800459i	0.999793	−	0.0203547i	\(-0.00647956\pi\)
							0.571196	+	0.820814i	\(0.306480\pi\)
\(17\)	6.00998	−	4.86679i	1.45763	−	1.18037i	0.510354	−	0.859965i	\(-0.329515\pi\)
							0.947281	−	0.320404i	\(-0.103819\pi\)
\(19\)	−0.112997	+	2.15611i	−0.0259233	+	0.494645i	0.954557	+	0.298028i	\(0.0963289\pi\)
							−0.980480	+	0.196617i	\(0.937004\pi\)
\(23\)	4.74100	+	1.00773i	0.988566	+	0.210126i	0.673696	−	0.739009i	\(-0.264706\pi\)
							0.314870	+	0.949135i	\(0.398039\pi\)
\(29\)	0.00373581	−	0.0235869i	0.000693722	−	0.00437999i	−0.987339	−	0.158624i	\(-0.949294\pi\)
							0.988033	+	0.154244i	\(0.0492942\pi\)
\(31\)	−0.305659	+	2.90815i	−0.0548979	+	0.522319i	0.932171	+	0.362020i	\(0.117913\pi\)
							−0.987068	+	0.160299i	\(0.948754\pi\)
\(37\)	−0.284536	−	2.70718i	−0.0467774	−	0.445057i	−0.992695	−	0.120651i	\(-0.961502\pi\)
							0.945918	−	0.324407i	\(-0.105165\pi\)
\(41\)	2.08835	−	6.05300i	0.326146	−	0.945320i
							\(43\)	−0.974186	+	0.316532i	−0.148562	+	0.0482707i	−0.382354	−	0.924016i	\(-0.624887\pi\)
0.233792	+	0.972287i	\(0.424887\pi\)
\(47\)	−2.80856	−	4.32480i	−0.409671	−	0.630837i	0.572261	−	0.820071i	\(-0.306066\pi\)
							−0.981932	+	0.189234i	\(0.939400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.11	✓	416
7.4	even	3	inner	287.2.bc.a.207.11	yes	416
41.21	even	20	inner	287.2.bc.a.226.11	yes	416
287.144	even	60	inner	287.2.bc.a.144.11	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.11	✓	416	1.1	even	1	trivial
287.2.bc.a.144.11	yes	416	287.144	even	60	inner
287.2.bc.a.207.11	yes	416	7.4	even	3	inner
287.2.bc.a.226.11	yes	416	41.21	even	20	inner