Embedded newform 287.3.o.a.139.6

• Label: 287.3.o.a.139.6
• Level: $287$
• Weight: $3$
• Character: 287.139
• 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.o (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			139.6
Character	\(\chi\)	\(=\)	287.139
Dual form			287.3.o.a.223.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08527 + 3.34011i) q^{2} +1.54191i q^{3} +(-6.74248 - 4.89870i) q^{4} +(3.83280 - 5.27539i) q^{5} +(-5.15016 - 1.67339i) q^{6} +(4.75633 + 5.13589i) q^{7} +(12.3145 - 8.94703i) q^{8} +6.62251 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 3 q^{7} - 2 q^{8} - 648 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08527	+	3.34011i	−0.542634	+	1.67006i	0.183915	+	0.982942i	\(0.441123\pi\)
							−0.726549	+	0.687115i	\(0.758877\pi\)
\(3\)			1.54191i			0.513971i	0.966415	+	0.256985i	\(0.0827292\pi\)
							−0.966415	+	0.256985i	\(0.917271\pi\)
\(5\)	3.83280	−	5.27539i	0.766559	−	1.05508i	−0.230081	−	0.973172i	\(-0.573899\pi\)
							0.996640	−	0.0819067i	\(-0.0261009\pi\)
\(7\)	4.75633	+	5.13589i	0.679475	+	0.733698i
							\(11\)	−11.6861	+	8.49046i	−1.06237	+	0.771860i	−0.974526	−	0.224274i	\(-0.927999\pi\)
−0.0878480	+	0.996134i	\(0.527999\pi\)
\(13\)	7.14291	+	2.32087i	0.549455	+	0.178529i	0.570571	−	0.821248i	\(-0.306722\pi\)
							−0.0211162	+	0.999777i	\(0.506722\pi\)
\(17\)	10.8233	+	14.8970i	0.636665	+	0.876294i	0.998432	−	0.0559745i	\(-0.0178265\pi\)
							−0.361767	+	0.932268i	\(0.617827\pi\)
\(19\)	−18.7080	+	6.07860i	−0.984633	+	0.319927i	−0.756709	−	0.653752i	\(-0.773194\pi\)
							−0.227924	+	0.973679i	\(0.573194\pi\)
\(23\)	1.46864	−	4.52002i	0.0638541	−	0.196523i	−0.914040	−	0.405625i	\(-0.867054\pi\)
							0.977894	+	0.209102i	\(0.0670540\pi\)
\(29\)	13.3379	+	9.69054i	0.459927	+	0.334156i	0.793503	−	0.608567i	\(-0.208255\pi\)
							−0.333576	+	0.942723i	\(0.608255\pi\)
\(31\)	31.3005	+	43.0814i	1.00969	+	1.38972i	0.919185	+	0.393826i	\(0.128849\pi\)
							0.0905073	+	0.995896i	\(0.471151\pi\)
\(37\)	−25.7283	−	18.6927i	−0.695360	−	0.505208i	0.183058	−	0.983102i	\(-0.441400\pi\)
							−0.878418	+	0.477894i	\(0.841400\pi\)
\(41\)	10.1800	−	39.7161i	0.248292	−	0.968685i
							\(43\)	1.98519	−	6.10977i	0.0461671	−	0.142088i	−0.925316	−	0.379197i	\(-0.876200\pi\)
0.971483	+	0.237110i	\(0.0762001\pi\)
\(47\)	−25.0240	−	8.13080i	−0.532426	−	0.172996i	0.0304510	−	0.999536i	\(-0.490306\pi\)
							−0.562877	+	0.826540i	\(0.690306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.o.a.139.6	yes	216
7.6	odd	2	inner	287.3.o.a.139.5	✓	216
41.18	even	5	inner	287.3.o.a.223.6	yes	216
287.223	odd	10	inner	287.3.o.a.223.5	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.o.a.139.5	✓	216	7.6	odd	2	inner
287.3.o.a.139.6	yes	216	1.1	even	1	trivial
287.3.o.a.223.5	yes	216	287.223	odd	10	inner
287.3.o.a.223.6	yes	216	41.18	even	5	inner