Embedded newform 287.3.q.a.73.1

• Label: 287.3.q.a.73.1
• 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			73.1
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.32140 + 1.91761i) q^{2} +(1.15997 - 0.310813i) q^{3} +(5.35446 - 9.27420i) q^{4} +(-2.42895 - 4.20707i) q^{5} +(-3.25670 + 3.25670i) q^{6} +(6.43721 - 2.74996i) q^{7} +25.7302i q^{8} +(-6.54530 + 3.77893i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.32140	+	1.91761i	−1.66070	+	0.958805i	−0.688318	+	0.725409i	\(0.741650\pi\)
							−0.972382	+	0.233396i	\(0.925016\pi\)
\(3\)	1.15997	−	0.310813i	0.386656	−	0.103604i	−0.0602538	−	0.998183i	\(-0.519191\pi\)
							0.446910	+	0.894579i	\(0.352524\pi\)
\(5\)	−2.42895	−	4.20707i	−0.485791	−	0.841415i	0.514076	−	0.857745i	\(-0.328135\pi\)
							−0.999867	+	0.0163302i	\(0.994802\pi\)
\(7\)	6.43721	−	2.74996i	0.919602	−	0.392852i
							\(11\)	0.864139	+	3.22501i	0.0785581	+	0.293183i	0.994017	−	0.109230i	\(-0.0348384\pi\)
−0.915458	+	0.402413i	\(0.868172\pi\)
\(13\)	11.4321	+	11.4321i	0.879395	+	0.879395i	0.993472	−	0.114077i	\(-0.0363911\pi\)
							−0.114077	+	0.993472i	\(0.536391\pi\)
\(17\)	−5.96592	−	22.2651i	−0.350937	−	1.30971i	−0.885522	−	0.464598i	\(-0.846199\pi\)
							0.534585	−	0.845115i	\(-0.320468\pi\)
\(19\)	22.0448	+	5.90688i	1.16025	+	0.310888i	0.787066	−	0.616869i	\(-0.211599\pi\)
							0.373185	+	0.927757i	\(0.378266\pi\)
\(23\)	−17.6024	−	30.4883i	−0.765324	−	1.32558i	−0.940075	−	0.340967i	\(-0.889246\pi\)
							0.174751	−	0.984613i	\(-0.444088\pi\)
\(29\)	12.8266	−	12.8266i	0.442296	−	0.442296i	−0.450487	−	0.892783i	\(-0.648750\pi\)
							0.892783	+	0.450487i	\(0.148750\pi\)
\(31\)	−17.6831	−	10.2093i	−0.570423	−	0.329334i	0.186895	−	0.982380i	\(-0.440157\pi\)
							−0.757318	+	0.653046i	\(0.773491\pi\)
\(37\)	1.91505	+	3.31697i	0.0517582	+	0.0896479i	0.890744	−	0.454506i	\(-0.150184\pi\)
							−0.838985	+	0.544154i	\(0.816851\pi\)
\(41\)	15.8075	−	37.8302i	0.385549	−	0.922687i
							\(43\)		−	73.6925i		−	1.71378i	−0.515499	−	0.856890i	\(-0.672394\pi\)
0.515499	−	0.856890i	\(-0.327606\pi\)
\(47\)	−4.35656	−	1.16734i	−0.0926928	−	0.0248370i	0.212175	−	0.977232i	\(-0.431945\pi\)
							−0.304867	+	0.952395i	\(0.598612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.1	✓	216
7.5	odd	6	inner	287.3.q.a.278.54	yes	216
41.9	even	4	inner	287.3.q.a.255.54	yes	216
287.173	odd	12	inner	287.3.q.a.173.1	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.1	✓	216	1.1	even	1	trivial
287.3.q.a.173.1	yes	216	287.173	odd	12	inner
287.3.q.a.255.54	yes	216	41.9	even	4	inner
287.3.q.a.278.54	yes	216	7.5	odd	6	inner