Embedded newform 287.2.w.a.3.7

• Label: 287.2.w.a.3.7
• Level: $287$
• Weight: $2$
• Character: 287.3
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58699 - 0.425234i) q^{2} +(0.715707 - 0.549181i) q^{3} +(0.605675 + 0.349687i) q^{4} +(1.24836 + 0.334498i) q^{5} +(-1.36935 + 0.567204i) q^{6} +(-0.971735 + 2.46084i) q^{7} +(1.51102 + 1.51102i) q^{8} +(-0.565821 + 2.11167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58699	−	0.425234i	−1.12217	−	0.300686i	−0.350411	−	0.936596i	\(-0.613958\pi\)
							−0.771763	+	0.635910i	\(0.780625\pi\)
\(3\)	0.715707	−	0.549181i	0.413213	−	0.317070i	−0.381149	−	0.924514i	\(-0.624472\pi\)
							0.794363	+	0.607444i	\(0.207805\pi\)
\(5\)	1.24836	+	0.334498i	0.558284	+	0.149592i	0.526918	−	0.849916i	\(-0.323347\pi\)
							0.0313663	+	0.999508i	\(0.490014\pi\)
\(7\)	−0.971735	+	2.46084i	−0.367281	+	0.930110i
							\(11\)	−2.29820	+	1.76347i	−0.692932	+	0.531705i	−0.894109	−	0.447850i	\(-0.852190\pi\)
0.201177	+	0.979555i	\(0.435523\pi\)
\(13\)	−0.312956	−	0.129631i	−0.0867983	−	0.0359530i	0.338862	−	0.940836i	\(-0.389958\pi\)
							−0.425660	+	0.904883i	\(0.639958\pi\)
\(17\)	−0.171173	+	1.30019i	−0.0415156	+	0.315343i	0.958069	+	0.286536i	\(0.0925039\pi\)
							−0.999585	+	0.0288062i	\(0.990829\pi\)
\(19\)	5.42796	+	4.16502i	1.24526	+	0.955522i	0.999903	−	0.0139112i	\(-0.00442822\pi\)
							0.245357	+	0.969433i	\(0.421095\pi\)
\(23\)	0.706206	−	0.407728i	0.147254	−	0.0850172i	−0.424563	−	0.905399i	\(-0.639572\pi\)
							0.571817	+	0.820381i	\(0.306239\pi\)
\(29\)	−2.14570	+	5.18019i	−0.398447	+	0.961937i	0.589587	+	0.807705i	\(0.299290\pi\)
							−0.988035	+	0.154232i	\(0.950710\pi\)
\(31\)	4.30081	−	7.44922i	0.772448	−	1.33792i	−0.163770	−	0.986499i	\(-0.552366\pi\)
							0.936218	−	0.351420i	\(-0.114301\pi\)
\(37\)	4.71225	+	8.16186i	0.774689	+	1.34180i	0.934969	+	0.354730i	\(0.115427\pi\)
							−0.160279	+	0.987072i	\(0.551240\pi\)
\(41\)	4.85533	+	4.17442i	0.758275	+	0.651935i
							\(43\)	7.77875	−	7.77875i	1.18625	−	1.18625i	0.208152	−	0.978097i	\(-0.433255\pi\)
0.978097	−	0.208152i	\(-0.0667448\pi\)
\(47\)	−1.74840	−	1.34160i	−0.255031	−	0.195692i	0.473331	−	0.880885i	\(-0.343051\pi\)
							−0.728362	+	0.685192i	\(0.759718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.7	✓	208
7.5	odd	6	inner	287.2.w.a.208.20	yes	208
41.14	odd	8	inner	287.2.w.a.178.20	yes	208
287.96	even	24	inner	287.2.w.a.96.7	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.7	✓	208	1.1	even	1	trivial
287.2.w.a.96.7	yes	208	287.96	even	24	inner
287.2.w.a.178.20	yes	208	41.14	odd	8	inner
287.2.w.a.208.20	yes	208	7.5	odd	6	inner