Embedded newform 287.3.q.a.73.13

• Label: 287.3.q.a.73.13
• 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			73.13
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.14335 + 1.23746i) q^{2} +(-3.28848 + 0.881145i) q^{3} +(1.06263 - 1.84052i) q^{4} +(-4.85635 - 8.41145i) q^{5} +(5.95797 - 5.95797i) q^{6} +(-2.38873 - 6.57982i) q^{7} -4.63986i q^{8} +(2.24345 - 1.29525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.14335	+	1.23746i	−1.07167	+	0.618731i	−0.928638	−	0.370986i	\(-0.879020\pi\)
							−0.143036	+	0.989718i	\(0.545686\pi\)
\(3\)	−3.28848	+	0.881145i	−1.09616	+	0.293715i	−0.761200	−	0.648517i	\(-0.775389\pi\)
							−0.334960	+	0.942232i	\(0.608723\pi\)
\(5\)	−4.85635	−	8.41145i	−0.971271	−	1.68229i	−0.691730	−	0.722156i	\(-0.743151\pi\)
							−0.279541	−	0.960134i	\(-0.590182\pi\)
\(7\)	−2.38873	−	6.57982i	−0.341247	−	0.939974i
							\(11\)	0.365997	+	1.36592i	0.0332725	+	0.124174i	0.980565	−	0.196197i	\(-0.0628591\pi\)
−0.947292	+	0.320371i	\(0.896192\pi\)
\(13\)	−8.76558	−	8.76558i	−0.674275	−	0.674275i	0.284424	−	0.958699i	\(-0.408198\pi\)
							−0.958699	+	0.284424i	\(0.908198\pi\)
\(17\)	−1.44622	−	5.39737i	−0.0850718	−	0.317492i	0.910256	−	0.414046i	\(-0.135885\pi\)
							−0.995328	+	0.0965537i	\(0.969218\pi\)
\(19\)	−5.68463	−	1.52319i	−0.299191	−	0.0801680i	0.106101	−	0.994355i	\(-0.466163\pi\)
							−0.405292	+	0.914187i	\(0.632830\pi\)
\(23\)	−19.8334	−	34.3524i	−0.862320	−	1.49358i	−0.869684	−	0.493609i	\(-0.835678\pi\)
							0.00736451	−	0.999973i	\(-0.497656\pi\)
\(29\)	15.8225	−	15.8225i	0.545602	−	0.545602i	−0.379563	−	0.925166i	\(-0.623926\pi\)
							0.925166	+	0.379563i	\(0.123926\pi\)
\(31\)	−35.6495	−	20.5823i	−1.14999	−	0.663944i	−0.201102	−	0.979570i	\(-0.564452\pi\)
							−0.948884	+	0.315626i	\(0.897786\pi\)
\(37\)	10.5574	+	18.2860i	0.285336	+	0.494216i	0.972691	−	0.232106i	\(-0.0745615\pi\)
							−0.687355	+	0.726322i	\(0.741228\pi\)
\(41\)	−36.5426	−	18.5914i	−0.891282	−	0.453450i
							\(43\)			5.96296i			0.138673i	0.997593	+	0.0693367i	\(0.0220883\pi\)
−0.997593	+	0.0693367i	\(0.977912\pi\)
\(47\)	45.6232	+	12.2247i	0.970705	+	0.260100i	0.709126	−	0.705082i	\(-0.249090\pi\)
							0.261580	+	0.965182i	\(0.415757\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.13	✓	216
7.5	odd	6	inner	287.3.q.a.278.42	yes	216
41.9	even	4	inner	287.3.q.a.255.42	yes	216
287.173	odd	12	inner	287.3.q.a.173.13	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.13	✓	216	1.1	even	1	trivial
287.3.q.a.173.13	yes	216	287.173	odd	12	inner
287.3.q.a.255.42	yes	216	41.9	even	4	inner
287.3.q.a.278.42	yes	216	7.5	odd	6	inner