Embedded newform 287.2.bb.a.6.2

• Label: 287.2.bb.a.6.2
• 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			6.2
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23448 + 2.42281i) q^{2} +(1.98219 - 0.821052i) q^{3} +(-3.17047 - 4.36378i) q^{4} +(-1.13770 - 0.180193i) q^{5} +(-0.457733 + 5.81604i) q^{6} +(1.07019 + 2.41965i) q^{7} +(9.11507 - 1.44369i) q^{8} +(1.13365 - 1.13365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23448	+	2.42281i	−0.872910	+	1.71318i	−0.191211	+	0.981549i	\(0.561241\pi\)
							−0.681699	+	0.731633i	\(0.738759\pi\)
\(3\)	1.98219	−	0.821052i	1.14442	−	0.474034i	0.271761	−	0.962365i	\(-0.412394\pi\)
							0.872659	+	0.488330i	\(0.162394\pi\)
\(5\)	−1.13770	−	0.180193i	−0.508793	−	0.0805849i	−0.103240	−	0.994656i	\(-0.532921\pi\)
							−0.405553	+	0.914072i	\(0.632921\pi\)
\(7\)	1.07019	+	2.41965i	0.404493	+	0.914541i
							\(11\)	3.20083	+	0.768450i	0.965085	+	0.231696i	0.685215	−	0.728341i	\(-0.259708\pi\)
0.279871	+	0.960038i	\(0.409708\pi\)
\(13\)	0.970201	+	1.13596i	0.269085	+	0.315059i	0.878453	−	0.477830i	\(-0.158576\pi\)
							−0.609367	+	0.792888i	\(0.708576\pi\)
\(17\)	2.92323	+	4.77028i	0.708987	+	1.15696i	0.980876	+	0.194634i	\(0.0623518\pi\)
							−0.271889	+	0.962329i	\(0.587648\pi\)
\(19\)	−0.919295	+	1.07636i	−0.210901	+	0.246933i	−0.855766	−	0.517363i	\(-0.826914\pi\)
							0.644865	+	0.764296i	\(0.276914\pi\)
\(23\)	5.07870	−	1.65017i	1.05898	−	0.344084i	0.272795	−	0.962072i	\(-0.412052\pi\)
							0.786188	+	0.617988i	\(0.212052\pi\)
\(29\)	3.55223	+	2.17681i	0.659632	+	0.404223i	0.811641	−	0.584156i	\(-0.198574\pi\)
							−0.152009	+	0.988379i	\(0.548574\pi\)
\(31\)	−7.42389	−	5.39378i	−1.33337	−	0.968751i	−0.999660	−	0.0260751i	\(-0.991699\pi\)
							−0.333711	−	0.942675i	\(-0.608301\pi\)
\(37\)	3.82563	−	2.77949i	0.628930	−	0.456945i	−0.227099	−	0.973872i	\(-0.572924\pi\)
							0.856029	+	0.516927i	\(0.172924\pi\)
\(41\)	2.95134	−	5.68240i	0.460921	−	0.887441i
							\(43\)	−1.32232	+	2.59519i	−0.201651	+	0.395763i	−0.969581	−	0.244771i	\(-0.921287\pi\)
0.767929	+	0.640534i	\(0.221287\pi\)
\(47\)	0.497964	−	6.32724i	0.0726356	−	0.922923i	−0.847366	−	0.531009i	\(-0.821813\pi\)
							0.920002	−	0.391914i	\(-0.128187\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.2	yes	416
7.6	odd	2	inner	287.2.bb.a.6.1	✓	416
41.7	odd	40	inner	287.2.bb.a.48.1	yes	416
287.48	even	40	inner	287.2.bb.a.48.2	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.1	✓	416	7.6	odd	2	inner
287.2.bb.a.6.2	yes	416	1.1	even	1	trivial
287.2.bb.a.48.1	yes	416	41.7	odd	40	inner
287.2.bb.a.48.2	yes	416	287.48	even	40	inner