Embedded newform 287.4.e.b.165.11

• Label: 287.4.e.b.165.11
• Level: $287$
• Weight: $4$
• Character: 287.165
• Analytic conductor: $16.934$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	287.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.9335481716\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			165.11
Character	\(\chi\)	\(=\)	287.165
Dual form			287.4.e.b.247.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59283 - 2.75886i) q^{2} +(-2.78174 + 4.81811i) q^{3} +(-1.07420 + 1.86057i) q^{4} +(-0.167371 - 0.289895i) q^{5} +17.7233 q^{6} +(-8.54008 - 16.4337i) q^{7} -18.6412 q^{8} +(-1.97614 - 3.42278i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 3 q^{2} + 6 q^{3} - 213 q^{4} - 4 q^{5} - 24 q^{6} - 30 q^{7} + 114 q^{8} - 452 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59283	−	2.75886i	−0.563150	−	0.975404i	−0.997219	−	0.0745243i	\(-0.976256\pi\)
							0.434070	−	0.900879i	\(-0.357077\pi\)
\(3\)	−2.78174	+	4.81811i	−0.535346	+	0.927246i	0.463801	+	0.885940i	\(0.346485\pi\)
							−0.999147	+	0.0413067i	\(0.986848\pi\)
\(5\)	−0.167371	−	0.289895i	−0.0149701	−	0.0259290i	0.858443	−	0.512908i	\(-0.171432\pi\)
							−0.873413	+	0.486979i	\(0.838099\pi\)
\(7\)	−8.54008	−	16.4337i	−0.461121	−	0.887337i
							\(11\)	19.2277	−	33.3034i	0.527034	−	0.912850i	−0.472469	−	0.881347i	\(-0.656637\pi\)
0.999504	−	0.0315030i	\(-0.0100294\pi\)
\(13\)	84.0040			1.79219			0.896097	−	0.443858i	\(-0.146391\pi\)
							0.896097	+	0.443858i	\(0.146391\pi\)
\(17\)	−47.5217	+	82.3101i	−0.677983	+	1.17430i	0.297604	+	0.954689i	\(0.403812\pi\)
							−0.975587	+	0.219612i	\(0.929521\pi\)
\(19\)	−15.8644	−	27.4780i	−0.191555	−	0.331783i	0.754211	−	0.656633i	\(-0.228020\pi\)
							−0.945766	+	0.324849i	\(0.894686\pi\)
\(23\)	−59.8939	−	103.739i	−0.542989	−	0.940485i	−0.998731	−	0.0503724i	\(-0.983959\pi\)
							0.455741	−	0.890112i	\(-0.349374\pi\)
\(29\)	−134.970			−0.864252			−0.432126	−	0.901813i	\(-0.642237\pi\)
							−0.432126	+	0.901813i	\(0.642237\pi\)
\(31\)	−160.358	+	277.748i	−0.929070	+	1.60920i	−0.144187	+	0.989550i	\(0.546057\pi\)
							−0.784882	+	0.619645i	\(0.787276\pi\)
\(37\)	−173.745	−	300.935i	−0.771987	−	1.33712i	−0.936473	−	0.350741i	\(-0.885930\pi\)
							0.164486	−	0.986379i	\(-0.447403\pi\)
\(41\)	−41.0000			−0.156174
							\(43\)	−253.641			−0.899531			−0.449765	−	0.893147i	\(-0.648492\pi\)
−0.449765	+	0.893147i	\(0.648492\pi\)
\(47\)	15.2640	+	26.4381i	0.0473720	+	0.0820508i	0.888739	−	0.458413i	\(-0.151582\pi\)
							−0.841367	+	0.540464i	\(0.818249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.4.e.b.165.11	✓	88
7.2	even	3	inner	287.4.e.b.247.11	yes	88
7.3	odd	6		2009.4.a.m.1.34		44
7.4	even	3		2009.4.a.l.1.34		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.b.165.11	✓	88	1.1	even	1	trivial
287.4.e.b.247.11	yes	88	7.2	even	3	inner
2009.4.a.l.1.34		44	7.4	even	3
2009.4.a.m.1.34		44	7.3	odd	6