Embedded newform 287.2.be.a.12.19

• Label: 287.2.be.a.12.19
• 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			12.19
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03626 + 0.839145i) q^{2} +(-2.26410 + 0.298075i) q^{3} +(-0.0461574 - 0.217153i) q^{4} +(-0.220901 - 0.0115769i) q^{5} +(-2.59632 - 1.59103i) q^{6} +(-2.51050 - 0.835088i) q^{7} +(1.34511 - 2.63992i) q^{8} +(2.13953 - 0.573285i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03626	+	0.839145i	0.732745	+	0.593365i	0.921371	−	0.388684i	\(-0.127070\pi\)
							−0.188626	+	0.982049i	\(0.560403\pi\)
\(3\)	−2.26410	+	0.298075i	−1.30718	+	0.172093i	−0.751738	−	0.659462i	\(-0.770784\pi\)
							−0.555442	+	0.831556i	\(0.687451\pi\)
\(5\)	−0.220901	−	0.0115769i	−0.0987900	−	0.00517736i	0.00287690	−	0.999996i	\(-0.499084\pi\)
							−0.101667	+	0.994818i	\(0.532418\pi\)
\(7\)	−2.51050	−	0.835088i	−0.948881	−	0.315633i
							\(11\)	−2.28287	−	4.78614i	−0.688312	−	1.44308i	−0.886413	−	0.462894i	\(-0.846811\pi\)
0.198102	−	0.980181i	\(-0.436522\pi\)
\(13\)	0.284290	+	1.18415i	0.0788479	+	0.328425i	0.997828	−	0.0658798i	\(-0.0209854\pi\)
							−0.918980	+	0.394305i	\(0.870985\pi\)
\(17\)	−0.0907221	−	0.0321264i	−0.0220033	−	0.00779179i	0.322782	−	0.946473i	\(-0.395382\pi\)
							−0.344786	+	0.938681i	\(0.612048\pi\)
\(19\)	−3.65633	−	3.46972i	−0.838819	−	0.796009i	0.142217	−	0.989836i	\(-0.454577\pi\)
							−0.981036	+	0.193826i	\(0.937910\pi\)
\(23\)	5.31475	−	0.558603i	1.10820	−	0.116477i	0.467300	−	0.884099i	\(-0.345227\pi\)
							0.640902	+	0.767622i	\(0.278560\pi\)
\(29\)	−2.19551	+	2.57061i	−0.407696	+	0.477351i	−0.925825	−	0.377951i	\(-0.876629\pi\)
							0.518129	+	0.855302i	\(0.326629\pi\)
\(31\)	−2.58873	+	2.87508i	−0.464950	+	0.516380i	−0.929327	−	0.369258i	\(-0.879612\pi\)
							0.464377	+	0.885638i	\(0.346278\pi\)
\(37\)	2.97398	+	3.30293i	0.488919	+	0.542999i	0.936234	−	0.351377i	\(-0.114287\pi\)
							−0.447315	+	0.894376i	\(0.647620\pi\)
\(41\)	−5.84735	+	2.60931i	−0.913203	+	0.407506i
							\(43\)	1.29608	−	8.18314i	0.197651	−	1.24792i	−0.666814	−	0.745224i	\(-0.732343\pi\)
0.864465	−	0.502693i	\(-0.167657\pi\)
\(47\)	−4.84037	+	2.62811i	−0.706041	+	0.383349i	−0.789989	−	0.613121i	\(-0.789914\pi\)
							0.0839479	+	0.996470i	\(0.473247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.19	✓	832
7.3	odd	6	inner	287.2.be.a.94.8	yes	832
41.24	odd	40	inner	287.2.be.a.229.8	yes	832
287.24	even	120	inner	287.2.be.a.24.19	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.19	✓	832	1.1	even	1	trivial
287.2.be.a.24.19	yes	832	287.24	even	120	inner
287.2.be.a.94.8	yes	832	7.3	odd	6	inner
287.2.be.a.229.8	yes	832	41.24	odd	40	inner