Embedded newform 287.3.q.a.73.4

• Label: 287.3.q.a.73.4
• 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			73.4
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.02057 + 1.74393i) q^{2} +(4.33367 - 1.16120i) q^{3} +(4.08257 - 7.07121i) q^{4} +(-3.12206 - 5.40756i) q^{5} +(-11.0651 + 11.0651i) q^{6} +(-6.98293 + 0.488488i) q^{7} +14.5274i q^{8} +(9.63808 - 5.56455i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.02057	+	1.74393i	−1.51029	+	0.871964i	−0.510358	+	0.859962i	\(0.670487\pi\)
							−0.999928	+	0.0120014i	\(0.996180\pi\)
\(3\)	4.33367	−	1.16120i	1.44456	−	0.387068i	0.550430	−	0.834881i	\(-0.314464\pi\)
							0.894127	+	0.447813i	\(0.147797\pi\)
\(5\)	−3.12206	−	5.40756i	−0.624411	−	1.08151i	−0.988654	−	0.150208i	\(-0.952006\pi\)
							0.364243	−	0.931304i	\(-0.381328\pi\)
\(7\)	−6.98293	+	0.488488i	−0.997562	+	0.0697840i
							\(11\)	1.41673	+	5.28730i	0.128793	+	0.480664i	0.999946	−	0.0103479i	\(-0.00329389\pi\)
−0.871153	+	0.491012i	\(0.836627\pi\)
\(13\)	−7.74911	−	7.74911i	−0.596086	−	0.596086i	0.343183	−	0.939269i	\(-0.388495\pi\)
							−0.939269	+	0.343183i	\(0.888495\pi\)
\(17\)	−1.07643	−	4.01729i	−0.0633194	−	0.236311i	0.927012	−	0.375031i	\(-0.122368\pi\)
							−0.990332	+	0.138720i	\(0.955701\pi\)
\(19\)	−12.4909	−	3.34692i	−0.657414	−	0.176154i	−0.0853353	−	0.996352i	\(-0.527196\pi\)
							−0.572079	+	0.820199i	\(0.693863\pi\)
\(23\)	11.0200	+	19.0873i	0.479132	+	0.829881i	0.999714	−	0.0239306i	\(-0.00761809\pi\)
							−0.520581	+	0.853812i	\(0.674285\pi\)
\(29\)	−31.3810	+	31.3810i	−1.08210	+	1.08210i	−0.0857914	+	0.996313i	\(0.527342\pi\)
							−0.996313	+	0.0857914i	\(0.972658\pi\)
\(31\)	−43.0132	−	24.8337i	−1.38752	−	0.801087i	−0.394488	−	0.918901i	\(-0.629078\pi\)
							−0.993036	+	0.117814i	\(0.962411\pi\)
\(37\)	−31.6453	−	54.8113i	−0.855278	−	1.48139i	−0.876387	−	0.481608i	\(-0.840053\pi\)
							0.0211084	−	0.999777i	\(-0.493280\pi\)
\(41\)	−40.8281	−	3.75092i	−0.995806	−	0.0914858i
							\(43\)		−	14.9195i		−	0.346965i	−0.984837	−	0.173482i	\(-0.944498\pi\)
0.984837	−	0.173482i	\(-0.0555020\pi\)
\(47\)	49.3614	+	13.2263i	1.05024	+	0.281412i	0.742352	−	0.670010i	\(-0.233710\pi\)
							0.307890	+	0.951422i	\(0.400377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.4	✓	216
7.5	odd	6	inner	287.3.q.a.278.51	yes	216
41.9	even	4	inner	287.3.q.a.255.51	yes	216
287.173	odd	12	inner	287.3.q.a.173.4	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.4	✓	216	1.1	even	1	trivial
287.3.q.a.173.4	yes	216	287.173	odd	12	inner
287.3.q.a.255.51	yes	216	41.9	even	4	inner
287.3.q.a.278.51	yes	216	7.5	odd	6	inner