Embedded newform 287.3.i.a.40.15

• Label: 287.3.i.a.40.15
• 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			40.15
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08643 + 1.88175i) q^{2} +(-2.10629 - 3.64821i) q^{3} +(-0.360668 - 0.624694i) q^{4} +(-6.23743 - 3.60118i) q^{5} +9.15338 q^{6} +(1.17747 - 6.90026i) q^{7} -7.12409 q^{8} +(-4.37295 + 7.57418i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08643	+	1.88175i	−0.543216	+	0.940877i	0.455501	+	0.890235i	\(0.349460\pi\)
							−0.998717	+	0.0506422i	\(0.983873\pi\)
\(3\)	−2.10629	−	3.64821i	−0.702098	−	1.21607i	−0.967729	−	0.251995i	\(-0.918913\pi\)
							0.265630	−	0.964075i	\(-0.414420\pi\)
\(5\)	−6.23743	−	3.60118i	−1.24749	−	0.720236i	−0.276878	−	0.960905i	\(-0.589300\pi\)
							−0.970607	+	0.240669i	\(0.922633\pi\)
\(7\)	1.17747	−	6.90026i	0.168209	−	0.985751i
							\(11\)	0.0645239	−	0.0372529i	0.00586581	−	0.00338663i	−0.497064	−	0.867714i	\(-0.665589\pi\)
0.502930	+	0.864327i	\(0.332255\pi\)
\(13\)	−7.23201			−0.556309			−0.278154	−	0.960536i	\(-0.589723\pi\)
							−0.278154	+	0.960536i	\(0.589723\pi\)
\(17\)	12.2548	+	21.2259i	0.720868	+	1.24858i	0.960652	+	0.277754i	\(0.0895900\pi\)
							−0.239784	+	0.970826i	\(0.577077\pi\)
\(19\)	7.58044	−	13.1297i	0.398971	−	0.691038i	−0.594628	−	0.804001i	\(-0.702701\pi\)
							0.993599	+	0.112963i	\(0.0360341\pi\)
\(23\)	−13.4472	+	23.2913i	−0.584663	+	1.01267i	0.410254	+	0.911971i	\(0.365440\pi\)
							−0.994917	+	0.100695i	\(0.967893\pi\)
\(29\)			24.5891i			0.847901i	0.905685	+	0.423950i	\(0.139357\pi\)
							−0.905685	+	0.423950i	\(0.860643\pi\)
\(31\)	22.3635	−	12.9116i	0.721405	−	0.416503i	−0.0938648	−	0.995585i	\(-0.529922\pi\)
							0.815269	+	0.579082i	\(0.196589\pi\)
\(37\)	−17.9417	+	31.0760i	−0.484911	+	0.839891i	−0.999850	−	0.0173360i	\(-0.994481\pi\)
							0.514938	+	0.857227i	\(0.327815\pi\)
\(41\)	14.7116	+	38.2697i	0.358818	+	0.933407i
							\(43\)	−51.4832			−1.19728			−0.598642	−	0.801017i	\(-0.704293\pi\)
−0.598642	+	0.801017i	\(0.704293\pi\)
\(47\)	17.6346	−	30.5441i	0.375205	−	0.649874i	−0.615153	−	0.788408i	\(-0.710906\pi\)
							0.990358	+	0.138534i	\(0.0442390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.15	✓	108
7.3	odd	6	inner	287.3.i.a.122.16	yes	108
41.40	even	2	inner	287.3.i.a.40.16	yes	108
287.122	odd	6	inner	287.3.i.a.122.15	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.15	✓	108	1.1	even	1	trivial
287.3.i.a.40.16	yes	108	41.40	even	2	inner
287.3.i.a.122.15	yes	108	287.122	odd	6	inner
287.3.i.a.122.16	yes	108	7.3	odd	6	inner