Embedded newform 287.2.bb.a.6.12

• Label: 287.2.bb.a.6.12
• 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			6.12
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.356263 + 0.699206i) q^{2} +(2.42868 - 1.00599i) q^{3} +(0.813605 + 1.11983i) q^{4} +(-2.63602 - 0.417504i) q^{5} +(-0.161854 + 2.05655i) q^{6} +(1.97176 - 1.76413i) q^{7} +(-2.62300 + 0.415443i) q^{8} +(2.76515 - 2.76515i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.356263	+	0.699206i	−0.251916	+	0.494413i	−0.981985	−	0.188961i	\(-0.939488\pi\)
							0.730069	+	0.683374i	\(0.239488\pi\)
\(3\)	2.42868	−	1.00599i	1.40220	−	0.580810i	0.451878	−	0.892080i	\(-0.350754\pi\)
							0.950321	+	0.311270i	\(0.100754\pi\)
\(5\)	−2.63602	−	0.417504i	−1.17886	−	0.186713i	−0.463907	−	0.885884i	\(-0.653553\pi\)
							−0.714955	+	0.699170i	\(0.753553\pi\)
\(7\)	1.97176	−	1.76413i	0.745255	−	0.666779i
							\(11\)	4.77369	+	1.14606i	1.43932	+	0.345551i	0.876650	−	0.481129i	\(-0.159773\pi\)
0.562673	+	0.826680i	\(0.309773\pi\)
\(13\)	1.45310	+	1.70136i	0.403018	+	0.471873i	0.924386	−	0.381459i	\(-0.124578\pi\)
							−0.521368	+	0.853332i	\(0.674578\pi\)
\(17\)	0.193424	+	0.315640i	0.0469123	+	0.0765539i	0.875237	−	0.483695i	\(-0.160706\pi\)
							−0.828324	+	0.560249i	\(0.810706\pi\)
\(19\)	1.63236	−	1.91125i	0.374490	−	0.438472i	−0.540871	−	0.841106i	\(-0.681905\pi\)
							0.915361	+	0.402634i	\(0.131905\pi\)
\(23\)	−5.02001	+	1.63110i	−1.04674	+	0.340108i	−0.781390	−	0.624043i	\(-0.785489\pi\)
							−0.265354	+	0.964151i	\(0.585489\pi\)
\(29\)	−5.75533	−	3.52687i	−1.06874	−	0.654924i	−0.127216	−	0.991875i	\(-0.540604\pi\)
							−0.941522	+	0.336952i	\(0.890604\pi\)
\(31\)	−4.74860	−	3.45006i	−0.852874	−	0.619649i	0.0730632	−	0.997327i	\(-0.476723\pi\)
							−0.925937	+	0.377678i	\(0.876723\pi\)
\(37\)	2.23628	−	1.62475i	0.367642	−	0.267108i	−0.388591	−	0.921411i	\(-0.627038\pi\)
							0.756232	+	0.654303i	\(0.227038\pi\)
\(41\)	−4.71923	+	4.32768i	−0.737021	+	0.675870i
							\(43\)	−1.99458	+	3.91459i	−0.304171	+	0.596969i	−0.991609	−	0.129274i	\(-0.958735\pi\)
0.687438	+	0.726243i	\(0.258735\pi\)
\(47\)	−0.636070	+	8.08203i	−0.0927803	+	1.17889i	0.757601	+	0.652718i	\(0.226371\pi\)
							−0.850381	+	0.526167i	\(0.823629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.12	yes	416
7.6	odd	2	inner	287.2.bb.a.6.11	✓	416
41.7	odd	40	inner	287.2.bb.a.48.11	yes	416
287.48	even	40	inner	287.2.bb.a.48.12	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.11	✓	416	7.6	odd	2	inner
287.2.bb.a.6.12	yes	416	1.1	even	1	trivial
287.2.bb.a.48.11	yes	416	41.7	odd	40	inner
287.2.bb.a.48.12	yes	416	287.48	even	40	inner