Embedded newform 287.3.m.a.85.7

• Label: 287.3.m.a.85.7
• Level: $287$
• Weight: $3$
• Character: 287.85
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(42\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			85.7
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.09220 + 2.09220i) q^{2} +(0.418806 - 1.01109i) q^{3} -4.75459i q^{4} +(-5.05847 - 5.05847i) q^{5} +(1.23917 + 2.99162i) q^{6} +(-1.01249 + 2.44436i) q^{7} +(1.57875 + 1.57875i) q^{8} +(5.51706 + 5.51706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.09220	+	2.09220i	−1.04610	+	1.04610i	−0.0472145	+	0.998885i	\(0.515034\pi\)
							−0.998885	+	0.0472145i	\(0.984966\pi\)
\(3\)	0.418806	−	1.01109i	0.139602	−	0.337029i	−0.838580	−	0.544778i	\(-0.816614\pi\)
							0.978182	+	0.207749i	\(0.0666138\pi\)
\(5\)	−5.05847	−	5.05847i	−1.01169	−	1.01169i	−0.999931	−	0.0117639i	\(-0.996255\pi\)
							−0.0117639	−	0.999931i	\(-0.503745\pi\)
\(7\)	−1.01249	+	2.44436i	−0.144641	+	0.349194i
							\(11\)	−16.8101	−	6.96296i	−1.52819	−	0.632996i	−0.548976	−	0.835838i	\(-0.684982\pi\)
−0.979211	+	0.202842i	\(0.934982\pi\)
\(13\)	5.18589	−	12.5198i	0.398915	−	0.963065i	−0.589009	−	0.808126i	\(-0.700482\pi\)
							0.987924	−	0.154939i	\(-0.0495181\pi\)
\(17\)	8.59203	+	20.7430i	0.505414	+	1.22018i	0.946498	+	0.322711i	\(0.104594\pi\)
							−0.441084	+	0.897466i	\(0.645406\pi\)
\(19\)	12.2017	+	29.4576i	0.642197	+	1.55040i	0.823710	+	0.567012i	\(0.191901\pi\)
							−0.181513	+	0.983389i	\(0.558099\pi\)
\(23\)			23.4760i			1.02070i	0.859968	+	0.510348i	\(0.170483\pi\)
							−0.859968	+	0.510348i	\(0.829517\pi\)
\(29\)	14.3895	−	34.7393i	0.496189	−	1.19791i	−0.455331	−	0.890322i	\(-0.650479\pi\)
							0.951521	−	0.307585i	\(-0.0995208\pi\)
\(31\)			27.0270i			0.871840i	0.899985	+	0.435920i	\(0.143577\pi\)
							−0.899985	+	0.435920i	\(0.856423\pi\)
\(37\)	10.8629			0.293592			0.146796	−	0.989167i	\(-0.453104\pi\)
							0.146796	+	0.989167i	\(0.453104\pi\)
\(41\)	2.44764	+	40.9269i	0.0596984	+	0.998216i
							\(43\)	−44.4340	+	44.4340i	−1.03335	+	1.03335i	−0.0339255	+	0.999424i	\(0.510801\pi\)
−0.999424	+	0.0339255i	\(0.989199\pi\)
\(47\)	−19.0337	−	45.9515i	−0.404973	−	0.977692i	−0.986440	−	0.164122i	\(-0.947521\pi\)
							0.581467	−	0.813570i	\(-0.302479\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.7	✓	168
41.14	odd	8	inner	287.3.m.a.260.7	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.7	✓	168	1.1	even	1	trivial
287.3.m.a.260.7	yes	168	41.14	odd	8	inner