Embedded newform 287.3.k.a.124.20

• Label: 287.3.k.a.124.20
• 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			124.20
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.721947 + 1.25045i) q^{2} +(3.30883 - 1.91036i) q^{3} +(0.957585 + 1.65859i) q^{4} +(-6.47182 - 3.73651i) q^{5} +5.51670i q^{6} +(0.469222 + 6.98426i) q^{7} -8.54088 q^{8} +(2.79891 - 4.84786i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.721947	+	1.25045i	−0.360973	+	0.625224i	−0.988121	−	0.153676i	\(-0.950889\pi\)
							0.627148	+	0.778900i	\(0.284222\pi\)
\(3\)	3.30883	−	1.91036i	1.10294	−	0.636785i	0.165951	−	0.986134i	\(-0.446931\pi\)
							0.936993	+	0.349349i	\(0.113597\pi\)
\(5\)	−6.47182	−	3.73651i	−1.29436	−	0.747302i	−0.314940	−	0.949112i	\(-0.601984\pi\)
							−0.979425	+	0.201810i	\(0.935318\pi\)
\(7\)	0.469222	+	6.98426i	0.0670317	+	0.997751i
							\(11\)	3.62996	+	6.28728i	0.329996	+	0.571571i	0.982511	−	0.186206i	\(-0.0596191\pi\)
−0.652514	+	0.757776i	\(0.726286\pi\)
\(13\)			15.9586i			1.22758i	0.789468	+	0.613792i	\(0.210357\pi\)
							−0.789468	+	0.613792i	\(0.789643\pi\)
\(17\)	−23.0933	+	13.3329i	−1.35843	+	0.784290i	−0.989412	−	0.145132i	\(-0.953639\pi\)
							−0.369018	+	0.929422i	\(0.620306\pi\)
\(19\)	27.6478	+	15.9625i	1.45515	+	0.840129i	0.998766	−	0.0496556i	\(-0.0158124\pi\)
							0.456380	+	0.889785i	\(0.349146\pi\)
\(23\)	9.51104	−	16.4736i	0.413524	−	0.716244i	−0.581749	−	0.813369i	\(-0.697631\pi\)
							0.995272	+	0.0971248i	\(0.0309646\pi\)
\(29\)	33.0530			1.13976			0.569879	−	0.821728i	\(-0.306990\pi\)
							0.569879	+	0.821728i	\(0.306990\pi\)
\(31\)	−3.65272	+	2.10890i	−0.117830	+	0.0680289i	−0.557756	−	0.830005i	\(-0.688338\pi\)
							0.439927	+	0.898034i	\(0.355004\pi\)
\(37\)	17.0485	−	29.5289i	0.460770	−	0.798078i	−0.538229	−	0.842799i	\(-0.680907\pi\)
							0.999000	+	0.0447208i	\(0.0142398\pi\)
\(41\)		−	6.40312i		−	0.156174i
							\(43\)	−39.3333			−0.914729			−0.457364	−	0.889279i	\(-0.651207\pi\)
−0.457364	+	0.889279i	\(0.651207\pi\)
\(47\)	−54.7831	−	31.6290i	−1.16560	−	0.672958i	−0.212959	−	0.977061i	\(-0.568310\pi\)
							−0.952639	+	0.304103i	\(0.901643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.20	✓	108
7.3	odd	6	inner	287.3.k.a.206.20	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.20	✓	108	1.1	even	1	trivial
287.3.k.a.206.20	yes	108	7.3	odd	6	inner