Embedded newform 287.3.i.a.40.7

• Label: 287.3.i.a.40.7
• Level: $287$
• Weight: $3$
• Character: 287.40
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			40.7
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55835 + 2.69914i) q^{2} +(-2.63338 - 4.56115i) q^{3} +(-2.85692 - 4.94833i) q^{4} +(4.74687 + 2.74061i) q^{5} +16.4149 q^{6} +(-6.99950 + 0.0840686i) q^{7} +5.34153 q^{8} +(-9.36940 + 16.2283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55835	+	2.69914i	−0.779176	+	1.34957i	0.153241	+	0.988189i	\(0.451029\pi\)
							−0.932417	+	0.361383i	\(0.882305\pi\)
\(3\)	−2.63338	−	4.56115i	−0.877794	−	1.52038i	−0.853756	−	0.520673i	\(-0.825681\pi\)
							−0.0240375	−	0.999711i	\(-0.507652\pi\)
\(5\)	4.74687	+	2.74061i	0.949374	+	0.548121i	0.892886	−	0.450282i	\(-0.148677\pi\)
							0.0564876	+	0.998403i	\(0.482010\pi\)
\(7\)	−6.99950	+	0.0840686i	−0.999928	+	0.0120098i
							\(11\)	−10.7410	+	6.20131i	−0.976454	+	0.563756i	−0.901198	−	0.433409i	\(-0.857311\pi\)
−0.0752560	+	0.997164i	\(0.523977\pi\)
\(13\)	17.1849			1.32192			0.660959	−	0.750422i	\(-0.270150\pi\)
							0.660959	+	0.750422i	\(0.270150\pi\)
\(17\)	4.40739	+	7.63383i	0.259259	+	0.449049i	0.966043	−	0.258380i	\(-0.0831885\pi\)
							−0.706785	+	0.707428i	\(0.749855\pi\)
\(19\)	16.6541	−	28.8458i	0.876532	−	1.51820i	0.0214112	−	0.999771i	\(-0.493184\pi\)
							0.855121	−	0.518428i	\(-0.173483\pi\)
\(23\)	−4.57393	+	7.92228i	−0.198866	+	0.344447i	−0.948161	−	0.317790i	\(-0.897059\pi\)
							0.749295	+	0.662237i	\(0.230393\pi\)
\(29\)			1.34146i			0.0462571i	0.999732	+	0.0231286i	\(0.00736270\pi\)
							−0.999732	+	0.0231286i	\(0.992637\pi\)
\(31\)	36.3907	−	21.0102i	1.17389	−	0.677748i	0.219300	−	0.975657i	\(-0.429623\pi\)
							0.954594	+	0.297909i	\(0.0962892\pi\)
\(37\)	−2.80360	+	4.85598i	−0.0757730	+	0.131243i	−0.901422	−	0.432941i	\(-0.857476\pi\)
							0.825649	+	0.564184i	\(0.190809\pi\)
\(41\)	3.11575	−	40.8814i	0.0759940	−	0.997108i
							\(43\)	65.6258			1.52618			0.763091	−	0.646292i	\(-0.223681\pi\)
0.763091	+	0.646292i	\(0.223681\pi\)
\(47\)	−19.2507	+	33.3432i	−0.409590	+	0.709430i	−0.994844	−	0.101420i	\(-0.967661\pi\)
							0.585254	+	0.810850i	\(0.300995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.7	✓	108
7.3	odd	6	inner	287.3.i.a.122.8	yes	108
41.40	even	2	inner	287.3.i.a.40.8	yes	108
287.122	odd	6	inner	287.3.i.a.122.7	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.7	✓	108	1.1	even	1	trivial
287.3.i.a.40.8	yes	108	41.40	even	2	inner
287.3.i.a.122.7	yes	108	287.122	odd	6	inner
287.3.i.a.122.8	yes	108	7.3	odd	6	inner