Embedded newform 287.2.bb.a.6.19

• Label: 287.2.bb.a.6.19
• 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			6.19
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.727703 - 1.42820i) q^{2} +(-1.25995 + 0.521890i) q^{3} +(-0.334625 - 0.460572i) q^{4} +(1.20688 + 0.191151i) q^{5} +(-0.171510 + 2.17924i) q^{6} +(1.62275 + 2.08966i) q^{7} +(2.26504 - 0.358747i) q^{8} +(-0.806204 + 0.806204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.727703	−	1.42820i	0.514564	−	1.00989i	−0.476833	−	0.878994i	\(-0.658215\pi\)
							0.991396	−	0.130894i	\(-0.0417847\pi\)
\(3\)	−1.25995	+	0.521890i	−0.727435	+	0.301313i	−0.715498	−	0.698615i	\(-0.753800\pi\)
							−0.0119376	+	0.999929i	\(0.503800\pi\)
\(5\)	1.20688	+	0.191151i	0.539734	+	0.0854855i	0.420349	−	0.907363i	\(-0.361908\pi\)
							0.119385	+	0.992848i	\(0.461908\pi\)
\(7\)	1.62275	+	2.08966i	0.613344	+	0.789816i
							\(11\)	0.406377	+	0.0975624i	0.122527	+	0.0294162i	0.294245	−	0.955730i	\(-0.404932\pi\)
−0.171718	+	0.985146i	\(0.554932\pi\)
\(13\)	0.969152	+	1.13473i	0.268794	+	0.314718i	0.878343	−	0.478031i	\(-0.158649\pi\)
							−0.609549	+	0.792749i	\(0.708649\pi\)
\(17\)	−0.588117	−	0.959719i	−0.142639	−	0.232766i	0.773340	−	0.633991i	\(-0.218584\pi\)
							−0.915979	+	0.401225i	\(0.868584\pi\)
\(19\)	2.89461	−	3.38916i	0.664070	−	0.777526i	−0.321479	−	0.946917i	\(-0.604180\pi\)
							0.985549	+	0.169391i	\(0.0541800\pi\)
\(23\)	1.64310	−	0.533876i	0.342610	−	0.111321i	−0.132658	−	0.991162i	\(-0.542351\pi\)
							0.475268	+	0.879841i	\(0.342351\pi\)
\(29\)	4.63396	+	2.83969i	0.860504	+	0.527318i	0.881349	−	0.472465i	\(-0.156636\pi\)
							−0.0208453	+	0.999783i	\(0.506636\pi\)
\(31\)	−5.32649	−	3.86992i	−0.956665	−	0.695058i	−0.00429126	−	0.999991i	\(-0.501366\pi\)
							−0.952374	+	0.304933i	\(0.901366\pi\)
\(37\)	−8.92531	+	6.48462i	−1.46731	+	1.06606i	−0.485931	+	0.873997i	\(0.661519\pi\)
							−0.981382	+	0.192068i	\(0.938481\pi\)
\(41\)	0.0895186	+	6.40250i	0.0139805	+	0.999902i
							\(43\)	−2.38194	+	4.67481i	−0.363242	+	0.712902i	−0.998221	−	0.0596273i	\(-0.981009\pi\)
0.634979	+	0.772529i	\(0.281009\pi\)
\(47\)	0.478528	−	6.08028i	0.0698005	−	0.886899i	−0.857958	−	0.513719i	\(-0.828267\pi\)
							0.927759	−	0.373180i	\(-0.121733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.19	✓	416
7.6	odd	2	inner	287.2.bb.a.6.20	yes	416
41.7	odd	40	inner	287.2.bb.a.48.20	yes	416
287.48	even	40	inner	287.2.bb.a.48.19	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.19	✓	416	1.1	even	1	trivial
287.2.bb.a.6.20	yes	416	7.6	odd	2	inner
287.2.bb.a.48.19	yes	416	287.48	even	40	inner
287.2.bb.a.48.20	yes	416	41.7	odd	40	inner