Embedded newform 287.3.i.a.40.3

• Label: 287.3.i.a.40.3
• 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			40.3
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79153 + 3.10301i) q^{2} +(-1.23268 - 2.13507i) q^{3} +(-4.41913 - 7.65416i) q^{4} +(-7.00807 - 4.04611i) q^{5} +8.83353 q^{6} +(-6.98958 - 0.381749i) q^{7} +17.3357 q^{8} +(1.46099 - 2.53050i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79153	+	3.10301i	−0.895763	+	1.55151i	−0.0629056	+	0.998019i	\(0.520037\pi\)
							−0.832857	+	0.553488i	\(0.813297\pi\)
\(3\)	−1.23268	−	2.13507i	−0.410894	−	0.711690i	0.584094	−	0.811686i	\(-0.301450\pi\)
							−0.994988	+	0.0999967i	\(0.968117\pi\)
\(5\)	−7.00807	−	4.04611i	−1.40161	−	0.809223i	−0.407056	−	0.913403i	\(-0.633445\pi\)
							−0.994558	+	0.104180i	\(0.966778\pi\)
\(7\)	−6.98958	−	0.381749i	−0.998512	−	0.0545356i
							\(11\)	−1.29153	+	0.745665i	−0.117412	+	0.0677878i	−0.557556	−	0.830139i	\(-0.688261\pi\)
0.440144	+	0.897927i	\(0.354927\pi\)
\(13\)	4.75178			0.365522			0.182761	−	0.983157i	\(-0.441497\pi\)
							0.182761	+	0.983157i	\(0.441497\pi\)
\(17\)	−13.6487	−	23.6403i	−0.802867	−	1.39061i	−0.917721	−	0.397225i	\(-0.869973\pi\)
							0.114854	−	0.993382i	\(-0.463360\pi\)
\(19\)	−8.71763	+	15.0994i	−0.458823	+	0.794704i	−0.998899	−	0.0469118i	\(-0.985062\pi\)
							0.540076	+	0.841616i	\(0.318395\pi\)
\(23\)	−3.81303	+	6.60436i	−0.165784	+	0.287146i	−0.936933	−	0.349508i	\(-0.886349\pi\)
							0.771150	+	0.636654i	\(0.219682\pi\)
\(29\)			35.5883i			1.22718i	0.789623	+	0.613592i	\(0.210276\pi\)
							−0.789623	+	0.613592i	\(0.789724\pi\)
\(31\)	−10.2977	+	5.94538i	−0.332184	+	0.191786i	−0.656810	−	0.754056i	\(-0.728095\pi\)
							0.324627	+	0.945842i	\(0.394761\pi\)
\(37\)	30.0319	−	52.0167i	0.811672	−	1.40586i	−0.100021	−	0.994985i	\(-0.531891\pi\)
							0.911693	−	0.410872i	\(-0.134776\pi\)
\(41\)	−37.5826	+	16.3875i	−0.916649	+	0.399694i
							\(43\)	53.0956			1.23478			0.617391	−	0.786657i	\(-0.288190\pi\)
0.617391	+	0.786657i	\(0.288190\pi\)
\(47\)	0.168213	−	0.291353i	0.00357900	−	0.00619900i	−0.864230	−	0.503096i	\(-0.832194\pi\)
							0.867809	+	0.496897i	\(0.165527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.3	✓	108
7.3	odd	6	inner	287.3.i.a.122.4	yes	108
41.40	even	2	inner	287.3.i.a.40.4	yes	108
287.122	odd	6	inner	287.3.i.a.122.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.3	✓	108	1.1	even	1	trivial
287.3.i.a.40.4	yes	108	41.40	even	2	inner
287.3.i.a.122.3	yes	108	287.122	odd	6	inner
287.3.i.a.122.4	yes	108	7.3	odd	6	inner