Embedded newform 287.2.bb.a.6.14

• Label: 287.2.bb.a.6.14
• Level: $287$
• Weight: $2$
• Character: 287.6
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.bb (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(26\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			6.14
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0362277 + 0.0711009i) q^{2} +(0.338380 - 0.140161i) q^{3} +(1.17183 + 1.61288i) q^{4} +(2.94779 + 0.466885i) q^{5} +(-0.00229312 + 0.0291368i) q^{6} +(1.65850 - 2.06140i) q^{7} +(-0.314762 + 0.0498534i) q^{8} +(-2.02646 + 2.02646i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0362277	+	0.0711009i	−0.0256169	+	0.0502759i	−0.903460	−	0.428672i	\(-0.858982\pi\)
							0.877843	+	0.478948i	\(0.158982\pi\)
\(3\)	0.338380	−	0.140161i	0.195363	−	0.0809222i	−0.282857	−	0.959162i	\(-0.591282\pi\)
							0.478220	+	0.878240i	\(0.341282\pi\)
\(5\)	2.94779	+	0.466885i	1.31829	+	0.208797i	0.775648	−	0.631165i	\(-0.217423\pi\)
							0.542645	+	0.839962i	\(0.317423\pi\)
\(7\)	1.65850	−	2.06140i	0.626856	−	0.779135i
							\(11\)	−2.51377	−	0.603502i	−0.757929	−	0.181963i	−0.163978	−	0.986464i	\(-0.552433\pi\)
−0.593951	+	0.804501i	\(0.702433\pi\)
\(13\)	−1.24804	−	1.46127i	−0.346145	−	0.405284i	0.559835	−	0.828604i	\(-0.310864\pi\)
							−0.905980	+	0.423320i	\(0.860864\pi\)
\(17\)	−1.06913	−	1.74465i	−0.259301	−	0.423141i	0.696324	−	0.717728i	\(-0.254818\pi\)
							−0.955625	+	0.294587i	\(0.904818\pi\)
\(19\)	1.51083	−	1.76895i	0.346608	−	0.405826i	−0.559529	−	0.828811i	\(-0.689018\pi\)
							0.906137	+	0.422985i	\(0.139018\pi\)
\(23\)	−0.937076	+	0.304474i	−0.195394	+	0.0634873i	−0.405079	−	0.914282i	\(-0.632756\pi\)
							0.209685	+	0.977769i	\(0.432756\pi\)
\(29\)	3.19866	+	1.96014i	0.593976	+	0.363989i	0.786766	−	0.617251i	\(-0.211754\pi\)
							−0.192790	+	0.981240i	\(0.561754\pi\)
\(31\)	−1.54584	−	1.12312i	−0.277641	−	0.201718i	0.440247	−	0.897877i	\(-0.354891\pi\)
							−0.717888	+	0.696159i	\(0.754891\pi\)
\(37\)	3.27053	−	2.37618i	0.537672	−	0.390641i	−0.285548	−	0.958364i	\(-0.592176\pi\)
							0.823220	+	0.567723i	\(0.192176\pi\)
\(41\)	−4.99444	+	4.00695i	−0.780000	+	0.625780i
							\(43\)	1.88337	−	3.69632i	0.287211	−	0.563684i	−0.701651	−	0.712521i	\(-0.747553\pi\)
0.988862	+	0.148838i	\(0.0475532\pi\)
\(47\)	0.472869	−	6.00837i	0.0689751	−	0.876411i	−0.860957	−	0.508679i	\(-0.830134\pi\)
							0.929932	−	0.367733i	\(-0.119866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.14	yes	416
7.6	odd	2	inner	287.2.bb.a.6.13	✓	416
41.7	odd	40	inner	287.2.bb.a.48.13	yes	416
287.48	even	40	inner	287.2.bb.a.48.14	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.13	✓	416	7.6	odd	2	inner
287.2.bb.a.6.14	yes	416	1.1	even	1	trivial
287.2.bb.a.48.13	yes	416	41.7	odd	40	inner
287.2.bb.a.48.14	yes	416	287.48	even	40	inner