Embedded newform 287.3.k.a.124.3

• Label: 287.3.k.a.124.3
• Level: $287$
• Weight: $3$
• Character: 287.124
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.k (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			124.3
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90578 + 3.30090i) q^{2} +(-0.0412365 + 0.0238079i) q^{3} +(-5.26396 - 9.11744i) q^{4} +(6.03383 + 3.48363i) q^{5} -0.181490i q^{6} +(-6.34884 - 2.94826i) q^{7} +24.8815 q^{8} +(-4.49887 + 7.79227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90578	+	3.30090i	−0.952888	+	1.65045i	−0.213758	+	0.976887i	\(0.568570\pi\)
							−0.739130	+	0.673563i	\(0.764763\pi\)
\(3\)	−0.0412365	+	0.0238079i	−0.0137455	+	0.00793598i	−0.506857	−	0.862030i	\(-0.669193\pi\)
							0.493111	+	0.869966i	\(0.335859\pi\)
\(5\)	6.03383	+	3.48363i	1.20677	+	0.696727i	0.962051	−	0.272868i	\(-0.0879723\pi\)
							0.244715	+	0.969595i	\(0.421306\pi\)
\(7\)	−6.34884	−	2.94826i	−0.906977	−	0.421180i
							\(11\)	−7.98777	−	13.8352i	−0.726161	−	1.25775i	−0.958494	−	0.285112i	\(-0.907969\pi\)
0.232333	−	0.972636i	\(-0.425364\pi\)
\(13\)		−	14.4159i		−	1.10891i	−0.832213	−	0.554456i	\(-0.812926\pi\)
							0.832213	−	0.554456i	\(-0.187074\pi\)
\(17\)	11.4416	−	6.60581i	0.673035	−	0.388577i	−0.124190	−	0.992258i	\(-0.539633\pi\)
							0.797226	+	0.603681i	\(0.206300\pi\)
\(19\)	−13.8634	−	8.00404i	−0.729653	−	0.421265i	0.0886424	−	0.996064i	\(-0.471747\pi\)
							−0.818295	+	0.574798i	\(0.805081\pi\)
\(23\)	12.0614	−	20.8909i	0.524408	−	0.908301i	−0.475188	−	0.879884i	\(-0.657620\pi\)
							0.999596	−	0.0284168i	\(-0.00904656\pi\)
\(29\)	−32.7084			−1.12788			−0.563939	−	0.825817i	\(-0.690715\pi\)
							−0.563939	+	0.825817i	\(0.690715\pi\)
\(31\)	43.9421	−	25.3700i	1.41749	−	0.818386i	0.421408	−	0.906871i	\(-0.361536\pi\)
							0.996077	+	0.0884854i	\(0.0282026\pi\)
\(37\)	−4.75606	+	8.23774i	−0.128542	+	0.222641i	−0.923112	−	0.384531i	\(-0.874363\pi\)
							0.794570	+	0.607173i	\(0.207696\pi\)
\(41\)		−	6.40312i		−	0.156174i
							\(43\)	−79.5463			−1.84991			−0.924957	−	0.380071i	\(-0.875900\pi\)
−0.924957	+	0.380071i	\(0.875900\pi\)
\(47\)	50.4230	+	29.1117i	1.07283	+	0.619398i	0.928953	−	0.370198i	\(-0.120710\pi\)
							0.143876	+	0.989596i	\(0.454043\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.3	✓	108
7.3	odd	6	inner	287.3.k.a.206.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.3	✓	108	1.1	even	1	trivial
287.3.k.a.206.3	yes	108	7.3	odd	6	inner