Embedded newform 287.3.p.a.146.9

• Label: 287.3.p.a.146.9
• Level: $287$
• Weight: $3$
• Character: 287.146
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.p (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			146.9
Character	\(\chi\)	\(=\)	287.146
Dual form			287.3.p.a.230.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.843522 - 2.59609i) q^{2} -5.83087 q^{3} +(-2.79211 + 2.02859i) q^{4} +(1.53896 + 2.11819i) q^{5} +(4.91847 + 15.1375i) q^{6} +(5.20818 - 4.67706i) q^{7} +(-1.21187 - 0.880473i) q^{8} +24.9991 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 5 q^{7} + 22 q^{8} + 552 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.843522	−	2.59609i	−0.421761	−	1.29805i	−0.906062	−	0.423146i	\(-0.860926\pi\)
							0.484300	−	0.874902i	\(-0.339074\pi\)
\(3\)	−5.83087			−1.94362			−0.971812	−	0.235756i	\(-0.924243\pi\)
							−0.971812	+	0.235756i	\(0.924243\pi\)
\(5\)	1.53896	+	2.11819i	0.307792	+	0.423639i	0.934691	−	0.355462i	\(-0.115676\pi\)
							−0.626899	+	0.779100i	\(0.715676\pi\)
\(7\)	5.20818	−	4.67706i	0.744026	−	0.668151i
							\(11\)	7.98874	−	10.9956i	0.726249	−	0.999596i	−0.273044	−	0.962002i	\(-0.588030\pi\)
0.999293	−	0.0375946i	\(-0.0119696\pi\)
\(13\)	1.98520	+	6.10983i	0.152708	+	0.469987i	0.997921	−	0.0644420i	\(-0.0205268\pi\)
							−0.845213	+	0.534429i	\(0.820527\pi\)
\(17\)	7.76125	+	5.63888i	0.456544	+	0.331699i	0.792174	−	0.610295i	\(-0.208949\pi\)
							−0.335630	+	0.941994i	\(0.608949\pi\)
\(19\)	1.46755	−	4.51666i	0.0772395	−	0.237719i	−0.904980	−	0.425454i	\(-0.860115\pi\)
							0.982220	+	0.187735i	\(0.0601146\pi\)
\(23\)	−6.84266	−	21.0595i	−0.297507	−	0.915632i	−0.982368	−	0.186959i	\(-0.940137\pi\)
							0.684861	−	0.728674i	\(-0.259863\pi\)
\(29\)	19.5351	+	26.8878i	0.673626	+	0.927166i	0.999836	−	0.0181310i	\(-0.00577160\pi\)
							−0.326210	+	0.945297i	\(0.605772\pi\)
\(31\)	10.0313	−	13.8070i	0.323592	−	0.445386i	−0.615968	−	0.787771i	\(-0.711235\pi\)
							0.939560	+	0.342385i	\(0.111235\pi\)
\(37\)	−35.6713	+	25.9167i	−0.964089	+	0.700452i	−0.954097	−	0.299498i	\(-0.903181\pi\)
							−0.00999255	+	0.999950i	\(0.503181\pi\)
\(41\)	−18.0846	−	36.7960i	−0.441088	−	0.897464i
							\(43\)	8.94636	+	27.5341i	0.208055	+	0.640327i	0.999574	+	0.0291828i	\(0.00929049\pi\)
−0.791519	+	0.611144i	\(0.790710\pi\)
\(47\)	−15.5361	−	47.8152i	−0.330555	−	1.01734i	−0.968870	−	0.247569i	\(-0.920368\pi\)
							0.638315	−	0.769775i	\(-0.279632\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.p.a.146.9	✓	216
7.6	odd	2	inner	287.3.p.a.146.10	yes	216
41.25	even	10	inner	287.3.p.a.230.10	yes	216
287.230	odd	10	inner	287.3.p.a.230.9	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.p.a.146.9	✓	216	1.1	even	1	trivial
287.3.p.a.146.10	yes	216	7.6	odd	2	inner
287.3.p.a.230.9	yes	216	287.230	odd	10	inner
287.3.p.a.230.10	yes	216	41.25	even	10	inner