Embedded newform 287.2.be.a.12.9

• Label: 287.2.be.a.12.9
• Level: $287$
• Weight: $2$
• Character: 287.12
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $832$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.be (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(832\)
Relative dimension:	\(26\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			12.9
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.833004 - 0.674554i) q^{2} +(-3.31166 + 0.435988i) q^{3} +(-0.176950 - 0.832483i) q^{4} +(3.11222 + 0.163105i) q^{5} +(3.05272 + 1.87071i) q^{6} +(-0.679716 - 2.55695i) q^{7} +(-1.38740 + 2.72292i) q^{8} +(7.87923 - 2.11123i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.833004	−	0.674554i	−0.589023	−	0.476981i	0.287868	−	0.957670i	\(-0.407053\pi\)
							−0.876891	+	0.480689i	\(0.840387\pi\)
\(3\)	−3.31166	+	0.435988i	−1.91199	+	0.251718i	−0.990586	−	0.136893i	\(-0.956288\pi\)
							−0.921402	+	0.388611i	\(0.872955\pi\)
\(5\)	3.11222	+	0.163105i	1.39183	+	0.0729426i	0.733450	−	0.679744i	\(-0.237909\pi\)
							0.658379	+	0.752687i	\(0.271242\pi\)
\(7\)	−0.679716	−	2.55695i	−0.256909	−	0.966436i
							\(11\)	0.767726	+	1.60957i	0.231478	+	0.485304i	0.985095	−	0.172010i	\(-0.0550261\pi\)
−0.753617	+	0.657314i	\(0.771693\pi\)
\(13\)	−0.656711	−	2.73540i	−0.182139	−	0.758662i	−0.986397	−	0.164382i	\(-0.947437\pi\)
							0.804258	−	0.594280i	\(-0.202563\pi\)
\(17\)	−5.83553	−	2.06647i	−1.41532	−	0.501193i	−0.487144	−	0.873322i	\(-0.661961\pi\)
							−0.928181	+	0.372129i	\(0.878628\pi\)
\(19\)	−3.20218	−	3.03875i	−0.734630	−	0.697138i	0.226261	−	0.974067i	\(-0.427350\pi\)
							−0.960890	+	0.276929i	\(0.910683\pi\)
\(23\)	−2.80330	+	0.294639i	−0.584529	+	0.0614365i	−0.392178	−	0.919890i	\(-0.628278\pi\)
							−0.192352	+	0.981326i	\(0.561611\pi\)
\(29\)	2.61756	−	3.06477i	0.486068	−	0.569113i	−0.462122	−	0.886816i	\(-0.652912\pi\)
							0.948190	+	0.317704i	\(0.102912\pi\)
\(31\)	−3.65775	+	4.06235i	−0.656952	+	0.729619i	−0.975916	−	0.218148i	\(-0.929998\pi\)
							0.318964	+	0.947767i	\(0.396665\pi\)
\(37\)	−2.25303	−	2.50224i	−0.370396	−	0.411366i	0.528916	−	0.848674i	\(-0.322599\pi\)
							−0.899312	+	0.437308i	\(0.855932\pi\)
\(41\)	−6.35961	+	0.745252i	−0.993204	+	0.116389i
							\(43\)	0.617411	−	3.89818i	0.0941544	−	0.594467i	−0.894825	−	0.446416i	\(-0.852700\pi\)
0.988980	−	0.148051i	\(-0.0472999\pi\)
\(47\)	8.00944	−	4.34877i	1.16830	−	0.634333i	0.227657	−	0.973741i	\(-0.426893\pi\)
							0.940639	+	0.339408i	\(0.110227\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.9	✓	832
7.3	odd	6	inner	287.2.be.a.94.18	yes	832
41.24	odd	40	inner	287.2.be.a.229.18	yes	832
287.24	even	120	inner	287.2.be.a.24.9	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.9	✓	832	1.1	even	1	trivial
287.2.be.a.24.9	yes	832	287.24	even	120	inner
287.2.be.a.94.18	yes	832	7.3	odd	6	inner
287.2.be.a.229.18	yes	832	41.24	odd	40	inner