Embedded newform 847.2.n.e.81.3

• Label: 847.2.n.e.81.3
• Level: $847$
• Weight: $2$
• Character: 847.81
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(3\) over \(\Q(\zeta_{15})\)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			81.3
Character	\(\chi\)	\(=\)	847.81
Dual form			847.2.n.e.366.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.260366 - 2.47722i) q^{2} +(-0.477450 - 0.530262i) q^{3} +(-4.11253 - 0.874144i) q^{4} +(2.01382 + 0.896611i) q^{5} +(-1.43789 + 1.04469i) q^{6} +(1.94692 - 1.79151i) q^{7} +(-1.69677 + 5.22212i) q^{8} +(0.260366 - 2.47722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.260366	−	2.47722i	0.184107	−	1.75166i	−0.379113	−	0.925351i	\(-0.623771\pi\)
							0.563219	−	0.826307i	\(-0.309563\pi\)
\(3\)	−0.477450	−	0.530262i	−0.275656	−	0.306147i	0.589381	−	0.807855i	\(-0.299372\pi\)
							−0.865037	+	0.501708i	\(0.832705\pi\)
\(5\)	2.01382	+	0.896611i	0.900608	+	0.400977i	0.804195	−	0.594365i	\(-0.202596\pi\)
							0.0964126	+	0.995341i	\(0.469263\pi\)
\(7\)	1.94692	−	1.79151i	0.735865	−	0.677128i
							\(11\)		0			0
\(13\)	2.65880	+	1.93173i	0.737419	+	0.535767i								0.891902	−	0.452229i	\(-0.149371\pi\)
							−0.154482	+	0.987996i	\(0.549371\pi\)
\(17\)	−0.155838	−	1.48270i	−0.0377962	−	0.359607i	−0.997031	−	0.0769974i	\(-0.975467\pi\)
							0.959235	−	0.282609i	\(-0.0912000\pi\)
\(19\)	6.76677	−	1.43832i	1.55240	−	0.329973i	0.649684	−	0.760205i	\(-0.274901\pi\)
							0.902719	+	0.430231i	\(0.141568\pi\)
\(23\)	−3.24543	−	5.62125i	−0.676719	−	1.17211i	−0.975963	−	0.217936i	\(-0.930068\pi\)
							0.299244	−	0.954177i	\(-0.403266\pi\)
\(29\)	−0.509801	−	1.56901i	−0.0946676	−	0.291357i	0.892499	−	0.451049i	\(-0.148950\pi\)
							−0.987167	+	0.159692i	\(0.948950\pi\)
\(31\)	−2.14706	+	0.955933i	−0.385623	+	0.171691i	−0.590386	−	0.807121i	\(-0.701024\pi\)
							0.204763	+	0.978812i	\(0.434358\pi\)
\(37\)	−3.71679	+	4.12791i	−0.611036	+	0.678624i	−0.966677	−	0.255998i	\(-0.917596\pi\)
							0.355641	+	0.934623i	\(0.384263\pi\)
\(41\)	−3.47642	+	10.6993i	−0.542925	+	1.67095i	0.182948	+	0.983123i	\(0.441436\pi\)
							−0.725873	+	0.687828i	\(0.758564\pi\)
\(43\)	5.26819			0.803391			0.401696	−	0.915773i	\(-0.368421\pi\)
							0.401696	+	0.915773i	\(0.368421\pi\)
\(47\)	−1.45828	+	0.309968i	−0.212713	+	0.0452135i	−0.313035	−	0.949741i	\(-0.601346\pi\)
							0.100323	+	0.994955i	\(0.468012\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.n.e.81.3		24
7.2	even	3	inner	847.2.n.e.807.1		24
11.2	odd	10		847.2.n.d.753.3		24
11.3	even	5	inner	847.2.n.e.487.1		24
11.4	even	5	inner	847.2.n.e.130.3		24
11.5	even	5		77.2.e.b.67.3	yes	6
11.6	odd	10		847.2.e.d.606.1		6
11.7	odd	10		847.2.n.d.130.1		24
11.8	odd	10		847.2.n.d.487.3		24
11.9	even	5	inner	847.2.n.e.753.1		24
11.10	odd	2		847.2.n.d.81.1		24
33.5	odd	10		693.2.i.g.298.1		6
44.27	odd	10		1232.2.q.k.529.2		6
77.2	odd	30		847.2.n.d.632.1		24
77.5	odd	30		539.2.e.l.177.3		6
77.9	even	15	inner	847.2.n.e.632.3		24
77.16	even	15		77.2.e.b.23.3	✓	6
77.17	even	30		5929.2.a.w.1.3		3
77.27	odd	10		539.2.e.l.67.3		6
77.30	odd	30		847.2.n.d.366.1		24
77.37	even	15	inner	847.2.n.e.9.1		24
77.38	odd	30		539.2.a.i.1.1		3
77.39	odd	30		5929.2.a.v.1.3		3
77.51	odd	30		847.2.n.d.9.3		24
77.58	even	15	inner	847.2.n.e.366.3		24
77.60	even	15		539.2.a.h.1.1		3
77.65	odd	6		847.2.n.d.807.3		24
77.72	odd	30		847.2.e.d.485.1		6
231.38	even	30		4851.2.a.bn.1.3		3
231.137	odd	30		4851.2.a.bo.1.3		3
231.170	odd	30		693.2.i.g.100.1		6
308.115	even	30		8624.2.a.ck.1.2		3
308.247	odd	30		1232.2.q.k.177.2		6
308.291	odd	30		8624.2.a.cl.1.2		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.3	✓	6	77.16	even	15
77.2.e.b.67.3	yes	6	11.5	even	5
539.2.a.h.1.1		3	77.60	even	15
539.2.a.i.1.1		3	77.38	odd	30
539.2.e.l.67.3		6	77.27	odd	10
539.2.e.l.177.3		6	77.5	odd	30
693.2.i.g.100.1		6	231.170	odd	30
693.2.i.g.298.1		6	33.5	odd	10
847.2.e.d.485.1		6	77.72	odd	30
847.2.e.d.606.1		6	11.6	odd	10
847.2.n.d.9.3		24	77.51	odd	30
847.2.n.d.81.1		24	11.10	odd	2
847.2.n.d.130.1		24	11.7	odd	10
847.2.n.d.366.1		24	77.30	odd	30
847.2.n.d.487.3		24	11.8	odd	10
847.2.n.d.632.1		24	77.2	odd	30
847.2.n.d.753.3		24	11.2	odd	10
847.2.n.d.807.3		24	77.65	odd	6
847.2.n.e.9.1		24	77.37	even	15	inner
847.2.n.e.81.3		24	1.1	even	1	trivial
847.2.n.e.130.3		24	11.4	even	5	inner
847.2.n.e.366.3		24	77.58	even	15	inner
847.2.n.e.487.1		24	11.3	even	5	inner
847.2.n.e.632.3		24	77.9	even	15	inner
847.2.n.e.753.1		24	11.9	even	5	inner
847.2.n.e.807.1		24	7.2	even	3	inner
1232.2.q.k.177.2		6	308.247	odd	30
1232.2.q.k.529.2		6	44.27	odd	10
4851.2.a.bn.1.3		3	231.38	even	30
4851.2.a.bo.1.3		3	231.137	odd	30
5929.2.a.v.1.3		3	77.39	odd	30
5929.2.a.w.1.3		3	77.17	even	30
8624.2.a.ck.1.2		3	308.115	even	30
8624.2.a.cl.1.2		3	308.291	odd	30