Embedded newform 287.3.p.a.146.13

• Label: 287.3.p.a.146.13
• Level: $287$
• Weight: $3$
• Character: 287.146
• 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.p (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			146.13
Character	\(\chi\)	\(=\)	287.146
Dual form			287.3.p.a.230.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707080 - 2.17617i) q^{2} -3.12138 q^{3} +(-0.999682 + 0.726311i) q^{4} +(4.21845 + 5.80620i) q^{5} +(2.20707 + 6.79266i) q^{6} +(-1.42526 + 6.85337i) q^{7} +(-5.11721 - 3.71787i) q^{8} +0.743042 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 5 q^{7} + 22 q^{8} + 552 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707080	−	2.17617i	−0.353540	−	1.08808i	−0.956851	−	0.290579i	\(-0.906152\pi\)
							0.603311	−	0.797506i	\(-0.293848\pi\)
\(3\)	−3.12138			−1.04046			−0.520231	−	0.854026i	\(-0.674154\pi\)
							−0.520231	+	0.854026i	\(0.674154\pi\)
\(5\)	4.21845	+	5.80620i	0.843690	+	1.16124i	0.985218	+	0.171306i	\(0.0547985\pi\)
							−0.141528	+	0.989934i	\(0.545201\pi\)
\(7\)	−1.42526	+	6.85337i	−0.203609	+	0.979052i
							\(11\)	4.73855	−	6.52206i	0.430778	−	0.592914i	−0.537354	−	0.843357i	\(-0.680576\pi\)
0.968131	+	0.250442i	\(0.0805761\pi\)
\(13\)	−5.27470	−	16.2338i	−0.405746	−	1.24876i	−0.920271	−	0.391282i	\(-0.872032\pi\)
							0.514525	−	0.857475i	\(-0.327968\pi\)
\(17\)	−11.4999	−	8.35520i	−0.676467	−	0.491482i	0.195717	−	0.980661i	\(-0.437297\pi\)
							−0.872184	+	0.489178i	\(0.837297\pi\)
\(19\)	7.42948	−	22.8656i	0.391025	−	1.20345i	−0.540989	−	0.841030i	\(-0.681950\pi\)
							0.932014	−	0.362422i	\(-0.118050\pi\)
\(23\)	−9.34359	−	28.7566i	−0.406243	−	1.25029i	−0.919853	−	0.392264i	\(-0.871692\pi\)
							0.513610	−	0.858024i	\(-0.328308\pi\)
\(29\)	−7.18636	−	9.89118i	−0.247806	−	0.341075i	0.666936	−	0.745115i	\(-0.267606\pi\)
							−0.914741	+	0.404040i	\(0.867606\pi\)
\(31\)	−20.0676	+	27.6206i	−0.647341	+	0.890989i	−0.998980	−	0.0451469i	\(-0.985624\pi\)
							0.351639	+	0.936136i	\(0.385624\pi\)
\(37\)	26.3352	−	19.1337i	0.711762	−	0.517126i	−0.171979	−	0.985101i	\(-0.555016\pi\)
							0.883742	+	0.467975i	\(0.155016\pi\)
\(41\)	−4.80798	+	40.7171i	−0.117268	+	0.993100i
							\(43\)	−17.5904	−	54.1377i	−0.409079	−	1.25902i	−0.917440	−	0.397873i	\(-0.869748\pi\)
0.508361	−	0.861144i	\(-0.330252\pi\)
\(47\)	−18.6462	−	57.3871i	−0.396728	−	1.22100i	−0.927608	−	0.373556i	\(-0.878138\pi\)
							0.530880	−	0.847447i	\(-0.321862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.p.a.146.13	✓	216
7.6	odd	2	inner	287.3.p.a.146.14	yes	216
41.25	even	10	inner	287.3.p.a.230.14	yes	216
287.230	odd	10	inner	287.3.p.a.230.13	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.p.a.146.13	✓	216	1.1	even	1	trivial
287.3.p.a.146.14	yes	216	7.6	odd	2	inner
287.3.p.a.230.13	yes	216	287.230	odd	10	inner
287.3.p.a.230.14	yes	216	41.25	even	10	inner