Embedded newform 287.3.x.a.31.11

• Label: 287.3.x.a.31.11
• Level: $287$
• Weight: $3$
• Character: 287.31
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.x (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			31.11
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.274542 + 2.61210i) q^{2} +(2.49203 - 4.31632i) q^{3} +(-2.83508 - 0.602615i) q^{4} +(6.74003 + 6.06875i) q^{5} +(10.5905 + 7.69444i) q^{6} +(6.28097 + 3.09022i) q^{7} +(-0.894076 + 2.75168i) q^{8} +(-7.92044 - 13.7186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.274542	+	2.61210i	−0.137271	+	1.30605i	0.681454	+	0.731861i	\(0.261348\pi\)
							−0.818725	+	0.574186i	\(0.805318\pi\)
\(3\)	2.49203	−	4.31632i	0.830677	−	1.43877i	−0.0668253	−	0.997765i	\(-0.521287\pi\)
							0.897502	−	0.441010i	\(-0.145380\pi\)
\(5\)	6.74003	+	6.06875i	1.34801	+	1.21375i	0.955851	+	0.293852i	\(0.0949371\pi\)
							0.392155	+	0.919899i	\(0.371730\pi\)
\(7\)	6.28097	+	3.09022i	0.897281	+	0.441460i
							\(11\)	−0.579962	+	0.522201i	−0.0527239	+	0.0474728i	−0.695071	−	0.718941i	\(-0.744627\pi\)
0.642348	+	0.766413i	\(0.277961\pi\)
\(13\)	4.30270	+	3.12610i	0.330977	+	0.240469i	0.740845	−	0.671676i	\(-0.234425\pi\)
							−0.409868	+	0.912145i	\(0.634425\pi\)
\(17\)	−21.3791	−	23.7439i	−1.25759	−	1.39670i	−0.882907	−	0.469548i	\(-0.844417\pi\)
							−0.374686	−	0.927152i	\(-0.622250\pi\)
\(19\)	−9.92724	−	4.41989i	−0.522486	−	0.232626i	0.128508	−	0.991708i	\(-0.458981\pi\)
							−0.650994	+	0.759083i	\(0.725648\pi\)
\(23\)	−1.56193	+	14.8608i	−0.0679099	+	0.646120i	0.906633	+	0.421920i	\(0.138644\pi\)
							−0.974543	+	0.224200i	\(0.928023\pi\)
\(29\)	30.6244	−	9.95047i	1.05601	−	0.343120i	0.270988	−	0.962583i	\(-0.412650\pi\)
							0.785026	+	0.619463i	\(0.212650\pi\)
\(31\)	−16.3962	+	14.7632i	−0.528908	+	0.476231i	−0.889783	−	0.456383i	\(-0.849145\pi\)
							0.360875	+	0.932614i	\(0.382478\pi\)
\(37\)	13.4829	−	14.9743i	0.364403	−	0.404711i	−0.532862	−	0.846202i	\(-0.678884\pi\)
							0.897266	+	0.441491i	\(0.145550\pi\)
\(41\)	−28.0700	−	29.8844i	−0.684634	−	0.728887i
							\(43\)	39.9150	+	29.0000i	0.928256	+	0.674417i	0.945565	−	0.325433i	\(-0.105510\pi\)
−0.0173092	+	0.999850i	\(0.505510\pi\)
\(47\)	4.63785	−	44.1262i	0.0986776	−	0.938855i	−0.827423	−	0.561579i	\(-0.810194\pi\)
							0.926101	−	0.377276i	\(-0.123139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.11	✓	432
7.5	odd	6	inner	287.3.x.a.236.44	yes	432
41.4	even	10	inner	287.3.x.a.45.44	yes	432
287.250	odd	30	inner	287.3.x.a.250.11	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.11	✓	432	1.1	even	1	trivial
287.3.x.a.45.44	yes	432	41.4	even	10	inner
287.3.x.a.236.44	yes	432	7.5	odd	6	inner
287.3.x.a.250.11	yes	432	287.250	odd	30	inner