Embedded newform 287.3.y.a.10.9

• Label: 287.3.y.a.10.9
• Level: $287$
• Weight: $3$
• Character: 287.10
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.y (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.9
Character	\(\chi\)	\(=\)	287.10
Dual form			287.3.y.a.201.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.288646 + 2.74628i) q^{2} +(-0.0116820 - 0.00674463i) q^{3} +(-3.54617 - 0.753762i) q^{4} +(4.04208 + 3.63950i) q^{5} +(0.0218947 - 0.0301354i) q^{6} +(-6.18024 - 3.28704i) q^{7} +(-0.319659 + 0.983810i) q^{8} +(-4.49991 - 7.79407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} - 24 q^{3} + 101 q^{4} - 9 q^{5} - 6 q^{7} - 16 q^{8} + 576 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.288646	+	2.74628i	−0.144323	+	1.37314i	0.647348	+	0.762195i	\(0.275878\pi\)
							−0.791671	+	0.610948i	\(0.790789\pi\)
\(3\)	−0.0116820	−	0.00674463i	−0.00389401	−	0.00224821i	0.498052	−	0.867147i	\(-0.334049\pi\)
							−0.501946	+	0.864899i	\(0.667382\pi\)
\(5\)	4.04208	+	3.63950i	0.808416	+	0.727901i	0.965702	−	0.259652i	\(-0.0836079\pi\)
							−0.157286	+	0.987553i	\(0.550275\pi\)
\(7\)	−6.18024	−	3.28704i	−0.882892	−	0.469577i
							\(11\)	−2.40318	−	2.66900i	−0.218471	−	0.242636i	0.623940	−	0.781472i	\(-0.285531\pi\)
−0.842411	+	0.538836i	\(0.818864\pi\)
\(13\)	−12.2320	+	16.8359i	−0.940921	+	1.29507i	0.0145234	+	0.999895i	\(0.495377\pi\)
							−0.955444	+	0.295172i	\(0.904623\pi\)
\(17\)	−5.39199	+	4.85497i	−0.317176	+	0.285586i	−0.812305	−	0.583233i	\(-0.801787\pi\)
							0.495129	+	0.868819i	\(0.335121\pi\)
\(19\)	−9.61050	+	21.5855i	−0.505816	+	1.13608i	0.462561	+	0.886588i	\(0.346931\pi\)
							−0.968377	+	0.249493i	\(0.919736\pi\)
\(23\)	−3.25994	+	31.0162i	−0.141736	+	1.34853i	0.660185	+	0.751103i	\(0.270478\pi\)
							−0.801921	+	0.597429i	\(0.796189\pi\)
\(29\)	−12.7147	−	39.1318i	−0.438437	−	1.34937i	−0.889523	−	0.456890i	\(-0.848963\pi\)
							0.451086	−	0.892481i	\(-0.351037\pi\)
\(31\)	27.7546	−	24.9904i	0.895310	−	0.806141i	−0.0864519	−	0.996256i	\(-0.527553\pi\)
							0.981762	+	0.190115i	\(0.0608862\pi\)
\(37\)	−36.4927	+	40.5292i	−0.986289	+	1.09538i	0.00914778	+	0.999958i	\(0.497088\pi\)
							−0.995436	+	0.0954264i	\(0.969579\pi\)
\(41\)	40.7232	−	4.75618i	0.993249	−	0.116004i
							\(43\)	51.9848	+	37.7692i	1.20895	+	0.878353i	0.995135	−	0.0985207i	\(-0.0314111\pi\)
0.213815	+	0.976874i	\(0.431411\pi\)
\(47\)	8.54747	+	0.898375i	0.181861	+	0.0191144i	0.195021	−	0.980799i	\(-0.437522\pi\)
							−0.0131601	+	0.999913i	\(0.504189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.y.a.10.9	✓	432
7.5	odd	6	inner	287.3.y.a.215.46	yes	432
41.37	even	5	inner	287.3.y.a.283.46	yes	432
287.201	odd	30	inner	287.3.y.a.201.9	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.y.a.10.9	✓	432	1.1	even	1	trivial
287.3.y.a.201.9	yes	432	287.201	odd	30	inner
287.3.y.a.215.46	yes	432	7.5	odd	6	inner
287.3.y.a.283.46	yes	432	41.37	even	5	inner