Embedded newform 287.3.p.a.146.8

• Label: 287.3.p.a.146.8
• 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			146.8
Character	\(\chi\)	\(=\)	287.146
Dual form			287.3.p.a.230.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.974724 - 2.99989i) q^{2} +2.12660 q^{3} +(-4.81319 + 3.49699i) q^{4} +(2.94240 + 4.04986i) q^{5} +(-2.07284 - 6.37956i) q^{6} +(-2.68944 + 6.46273i) q^{7} +(4.97468 + 3.61432i) q^{8} -4.47759 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 5 q^{7} + 22 q^{8} + 552 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.974724	−	2.99989i	−0.487362	−	1.49995i	−0.828531	−	0.559944i	\(-0.810823\pi\)
							0.341169	−	0.940002i	\(-0.389177\pi\)
\(3\)	2.12660			0.708865			0.354433	−	0.935082i	\(-0.384674\pi\)
							0.354433	+	0.935082i	\(0.384674\pi\)
\(5\)	2.94240	+	4.04986i	0.588479	+	0.809972i	0.994593	−	0.103850i	\(-0.0331163\pi\)
							−0.406114	+	0.913823i	\(0.633116\pi\)
\(7\)	−2.68944	+	6.46273i	−0.384206	+	0.923247i
							\(11\)	−6.11374	+	8.41485i	−0.555795	+	0.764986i	−0.990784	−	0.135450i	\(-0.956752\pi\)
0.434989	+	0.900436i	\(0.356752\pi\)
\(13\)	1.51016	+	4.64779i	0.116166	+	0.357522i	0.992188	−	0.124748i	\(-0.0398123\pi\)
							−0.876022	+	0.482270i	\(0.839812\pi\)
\(17\)	1.86749	+	1.35681i	0.109852	+	0.0798125i	0.641355	−	0.767244i	\(-0.278372\pi\)
							−0.531503	+	0.847057i	\(0.678372\pi\)
\(19\)	−5.26775	+	16.2125i	−0.277250	+	0.853287i	0.711366	+	0.702822i	\(0.248077\pi\)
							−0.988615	+	0.150465i	\(0.951923\pi\)
\(23\)	2.44765	+	7.53310i	0.106420	+	0.327526i	0.990061	−	0.140638i	\(-0.0449155\pi\)
							−0.883641	+	0.468164i	\(0.844915\pi\)
\(29\)	25.7471	+	35.4379i	0.887831	+	1.22200i	0.974190	+	0.225731i	\(0.0724771\pi\)
							−0.0863582	+	0.996264i	\(0.527523\pi\)
\(31\)	3.13200	−	4.31082i	0.101032	−	0.139059i	−0.755508	−	0.655140i	\(-0.772610\pi\)
							0.856540	+	0.516081i	\(0.172610\pi\)
\(37\)	−1.32734	+	0.964368i	−0.0358740	+	0.0260640i	−0.605578	−	0.795786i	\(-0.707058\pi\)
							0.569704	+	0.821850i	\(0.307058\pi\)
\(41\)	−10.3941	−	39.6606i	−0.253514	−	0.967332i
							\(43\)	−24.1947	−	74.4635i	−0.562667	−	1.73171i	−0.674785	−	0.738015i	\(-0.735763\pi\)
0.112118	−	0.993695i	\(-0.464237\pi\)
\(47\)	1.26259	+	3.88586i	0.0268636	+	0.0826778i	0.963589	−	0.267386i	\(-0.0861599\pi\)
							−0.936726	+	0.350064i	\(0.886160\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.p.a.146.8	yes	216
7.6	odd	2	inner	287.3.p.a.146.7	✓	216
41.25	even	10	inner	287.3.p.a.230.7	yes	216
287.230	odd	10	inner	287.3.p.a.230.8	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.p.a.146.7	✓	216	7.6	odd	2	inner
287.3.p.a.146.8	yes	216	1.1	even	1	trivial
287.3.p.a.230.7	yes	216	41.25	even	10	inner
287.3.p.a.230.8	yes	216	287.230	odd	10	inner