Embedded newform 287.3.i.a.40.18

• Label: 287.3.i.a.40.18
• 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			40.18
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.736544 + 1.27573i) q^{2} +(1.76416 + 3.05561i) q^{3} +(0.915005 + 1.58483i) q^{4} +(2.30976 + 1.33354i) q^{5} -5.19752 q^{6} +(1.63087 + 6.80737i) q^{7} -8.58812 q^{8} +(-1.72450 + 2.98691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.736544	+	1.27573i	−0.368272	+	0.637866i	−0.989296	−	0.145926i	\(-0.953384\pi\)
							0.621023	+	0.783792i	\(0.286717\pi\)
\(3\)	1.76416	+	3.05561i	0.588052	+	1.01854i	0.994487	+	0.104857i	\(0.0334384\pi\)
							−0.406435	+	0.913680i	\(0.633228\pi\)
\(5\)	2.30976	+	1.33354i	0.461952	+	0.266708i	0.712865	−	0.701302i	\(-0.247397\pi\)
							−0.250913	+	0.968010i	\(0.580731\pi\)
\(7\)	1.63087	+	6.80737i	0.232982	+	0.972481i
							\(11\)	1.94896	−	1.12523i	0.177178	−	0.102294i	−0.408788	−	0.912629i	\(-0.634049\pi\)
0.585966	+	0.810336i	\(0.300715\pi\)
\(13\)	−5.45408			−0.419545			−0.209772	−	0.977750i	\(-0.567272\pi\)
							−0.209772	+	0.977750i	\(0.567272\pi\)
\(17\)	9.00362	+	15.5947i	0.529624	+	0.917336i	0.999403	+	0.0345521i	\(0.0110005\pi\)
							−0.469778	+	0.882784i	\(0.655666\pi\)
\(19\)	11.3090	−	19.5878i	0.595212	−	1.03094i	−0.398305	−	0.917253i	\(-0.630402\pi\)
							0.993517	−	0.113684i	\(-0.0362651\pi\)
\(23\)	10.5649	−	18.2989i	0.459342	−	0.795604i	−0.539584	−	0.841932i	\(-0.681418\pi\)
							0.998926	+	0.0463274i	\(0.0147517\pi\)
\(29\)			51.2608i			1.76761i	0.467853	+	0.883806i	\(0.345028\pi\)
							−0.467853	+	0.883806i	\(0.654972\pi\)
\(31\)	−1.50289	+	0.867694i	−0.0484803	+	0.0279901i	−0.524044	−	0.851691i	\(-0.675577\pi\)
							0.475564	+	0.879681i	\(0.342244\pi\)
\(37\)	28.2743	−	48.9726i	0.764171	−	1.32358i	−0.176513	−	0.984298i	\(-0.556482\pi\)
							0.940684	−	0.339284i	\(-0.110185\pi\)
\(41\)	−25.7217	−	31.9280i	−0.627358	−	0.778731i
							\(43\)	−42.8482			−0.996469			−0.498235	−	0.867042i	\(-0.666018\pi\)
−0.498235	+	0.867042i	\(0.666018\pi\)
\(47\)	0.314847	−	0.545332i	0.00669888	−	0.0116028i	−0.862657	−	0.505790i	\(-0.831201\pi\)
							0.869355	+	0.494187i	\(0.164534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.18	yes	108
7.3	odd	6	inner	287.3.i.a.122.17	yes	108
41.40	even	2	inner	287.3.i.a.40.17	✓	108
287.122	odd	6	inner	287.3.i.a.122.18	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.17	✓	108	41.40	even	2	inner
287.3.i.a.40.18	yes	108	1.1	even	1	trivial
287.3.i.a.122.17	yes	108	7.3	odd	6	inner
287.3.i.a.122.18	yes	108	287.122	odd	6	inner