Embedded newform 287.3.ba.a.15.4

• Label: 287.3.ba.a.15.4
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.4
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.58163 - 0.567275i) q^{2} +(-4.62584 + 1.91609i) q^{3} +(8.70208 + 2.82748i) q^{4} +(1.82688 + 3.58546i) q^{5} +(17.6550 - 4.23860i) q^{6} +(2.57265 + 0.617638i) q^{7} +(-16.6395 - 8.47827i) q^{8} +(11.3631 - 11.3631i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.58163	−	0.567275i	−1.79082	−	0.283638i	−0.829384	−	0.558678i	\(-0.811309\pi\)
							−0.961433	+	0.275041i	\(0.911309\pi\)
\(3\)	−4.62584	+	1.91609i	−1.54195	+	0.638696i	−0.981838	−	0.189722i	\(-0.939241\pi\)
							−0.560110	+	0.828418i	\(0.689241\pi\)
\(5\)	1.82688	+	3.58546i	0.365376	+	0.717091i	0.998370	−	0.0570654i	\(-0.0181744\pi\)
							−0.632994	+	0.774156i	\(0.718174\pi\)
\(7\)	2.57265	+	0.617638i	0.367521	+	0.0882341i
							\(11\)	1.57817	−	1.34789i	0.143470	−	0.122535i	−0.574816	−	0.818282i	\(-0.694927\pi\)
0.718287	+	0.695747i	\(0.244927\pi\)
\(13\)	1.43635	−	2.34392i	0.110489	−	0.180301i	−0.792617	−	0.609720i	\(-0.791282\pi\)
							0.903105	+	0.429419i	\(0.141282\pi\)
\(17\)	−1.57594	+	20.0242i	−0.0927024	+	1.17790i	0.758003	+	0.652252i	\(0.226175\pi\)
							−0.850705	+	0.525644i	\(0.823825\pi\)
\(19\)	−18.4166	−	30.0531i	−0.969293	−	1.58174i	−0.803265	−	0.595621i	\(-0.796906\pi\)
							−0.166028	−	0.986121i	\(-0.553094\pi\)
\(23\)	−6.92561	−	9.53228i	−0.301113	−	0.414447i	0.631471	−	0.775400i	\(-0.282452\pi\)
							−0.932584	+	0.360953i	\(0.882452\pi\)
\(29\)	−2.00435	−	25.4677i	−0.0691157	−	0.878198i	−0.929564	−	0.368660i	\(-0.879817\pi\)
							0.860449	−	0.509537i	\(-0.170183\pi\)
\(31\)	47.3695	−	15.3913i	1.52805	−	0.496493i	0.579999	−	0.814618i	\(-0.303053\pi\)
							0.948049	+	0.318125i	\(0.103053\pi\)
\(37\)	8.53505	−	26.2682i	0.230677	−	0.709950i	−0.766989	−	0.641661i	\(-0.778246\pi\)
							0.997666	−	0.0682898i	\(-0.0217542\pi\)
\(41\)	37.3005	+	17.0198i	0.909768	+	0.415117i
							\(43\)	42.5920	+	6.74592i	0.990513	+	0.156882i	0.630603	−	0.776105i	\(-0.282808\pi\)
0.359910	+	0.932987i	\(0.382808\pi\)
\(47\)	−88.8372	+	21.3279i	−1.89015	+	0.453786i	−0.999998	−	0.00187256i	\(-0.999404\pi\)
							−0.890155	+	0.455658i	\(0.849404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.4	✓	672
41.11	odd	40	inner	287.3.ba.a.134.4	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.4	✓	672	1.1	even	1	trivial
287.3.ba.a.134.4	yes	672	41.11	odd	40	inner