Embedded newform 287.3.q.a.73.15

• Label: 287.3.q.a.73.15
• 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			73.15
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69267 + 0.977266i) q^{2} +(1.83287 - 0.491117i) q^{3} +(-0.0899034 + 0.155717i) q^{4} +(-2.23127 - 3.86467i) q^{5} +(-2.62250 + 2.62250i) q^{6} +(-4.95227 - 4.94722i) q^{7} -8.16956i q^{8} +(-4.67600 + 2.69969i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69267	+	0.977266i	−0.846337	+	0.488633i	−0.859413	−	0.511282i	\(-0.829171\pi\)
							0.0130763	+	0.999915i	\(0.495838\pi\)
\(3\)	1.83287	−	0.491117i	0.610958	−	0.163706i	0.0599438	−	0.998202i	\(-0.480908\pi\)
							0.551014	+	0.834496i	\(0.314241\pi\)
\(5\)	−2.23127	−	3.86467i	−0.446254	−	0.772934i	0.551885	−	0.833920i	\(-0.313909\pi\)
							−0.998139	+	0.0609863i	\(0.980575\pi\)
\(7\)	−4.95227	−	4.94722i	−0.707468	−	0.706746i
							\(11\)	2.28687	+	8.53473i	0.207898	+	0.775884i	0.988547	+	0.150915i	\(0.0482219\pi\)
−0.780649	+	0.624970i	\(0.785111\pi\)
\(13\)	18.2277	+	18.2277i	1.40213	+	1.40213i	0.793291	+	0.608843i	\(0.208366\pi\)
							0.608843	+	0.793291i	\(0.291634\pi\)
\(17\)	4.38956	+	16.3821i	0.258209	+	0.963651i	0.966277	+	0.257506i	\(0.0829006\pi\)
							−0.708067	+	0.706145i	\(0.750433\pi\)
\(19\)	8.30079	+	2.22419i	0.436884	+	0.117063i	0.470556	−	0.882370i	\(-0.344053\pi\)
							−0.0336719	+	0.999433i	\(0.510720\pi\)
\(23\)	5.68923	+	9.85403i	0.247358	+	0.428436i	0.962792	−	0.270244i	\(-0.0871044\pi\)
							−0.715434	+	0.698680i	\(0.753771\pi\)
\(29\)	−0.313587	+	0.313587i	−0.0108134	+	0.0108134i	−0.712493	−	0.701679i	\(-0.752434\pi\)
							0.701679	+	0.712493i	\(0.252434\pi\)
\(31\)	45.2988	+	26.1533i	1.46125	+	0.843654i	0.999069	−	0.0431305i	\(-0.0137331\pi\)
							0.462183	+	0.886785i	\(0.347066\pi\)
\(37\)	−20.3917	−	35.3195i	−0.551128	−	0.954582i	−0.998194	−	0.0600808i	\(-0.980864\pi\)
							0.447065	−	0.894501i	\(-0.352469\pi\)
\(41\)	5.84868	−	40.5807i	0.142651	−	0.989773i
							\(43\)			66.2123i			1.53982i	0.638152	+	0.769910i	\(0.279699\pi\)
−0.638152	+	0.769910i	\(0.720301\pi\)
\(47\)	−33.8895	−	9.08065i	−0.721052	−	0.193205i	−0.120411	−	0.992724i	\(-0.538421\pi\)
							−0.600641	+	0.799519i	\(0.705088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.15	✓	216
7.5	odd	6	inner	287.3.q.a.278.40	yes	216
41.9	even	4	inner	287.3.q.a.255.40	yes	216
287.173	odd	12	inner	287.3.q.a.173.15	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.15	✓	216	1.1	even	1	trivial
287.3.q.a.173.15	yes	216	287.173	odd	12	inner
287.3.q.a.255.40	yes	216	41.9	even	4	inner
287.3.q.a.278.40	yes	216	7.5	odd	6	inner