Embedded newform 847.2.n.d.81.2

• Label: 847.2.n.d.81.2
• 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			81.2
Character	\(\chi\)	\(=\)	847.81
Dual form			847.2.n.d.366.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0686355 - 0.653023i) q^{2} +(1.27953 + 1.42106i) q^{3} +(1.53457 + 0.326182i) q^{4} +(-3.26031 - 1.45158i) q^{5} +(1.01581 - 0.738028i) q^{6} +(-2.40923 - 1.09345i) q^{7} +(0.724144 - 2.22869i) q^{8} +(-0.0686355 + 0.653023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0686355	−	0.653023i	0.0485326	−	0.461757i	−0.943085	−	0.332551i	\(-0.892091\pi\)
							0.991618	−	0.129206i	\(-0.0412428\pi\)
\(3\)	1.27953	+	1.42106i	0.738738	+	0.820451i	0.989029	−	0.147719i	\(-0.0471932\pi\)
							−0.250292	+	0.968170i	\(0.580527\pi\)
\(5\)	−3.26031	−	1.45158i	−1.45805	−	0.649167i	−0.483914	−	0.875116i	\(-0.660785\pi\)
							−0.974139	+	0.225948i	\(0.927452\pi\)
\(7\)	−2.40923	−	1.09345i	−0.910602	−	0.413285i
							\(11\)		0			0
\(13\)	−4.78309	−	3.47512i	−1.32659	−	0.963825i								−0.999825	−	0.0187202i	\(-0.994041\pi\)
							−0.326767	−	0.945105i	\(-0.605959\pi\)
\(17\)	−0.173164	−	1.64755i	−0.0419984	−	0.399588i	−0.995246	−	0.0973968i	\(-0.968948\pi\)
							0.953247	−	0.302192i	\(-0.0977183\pi\)
\(19\)	1.44871	−	0.307934i	0.332358	−	0.0706448i	−0.0387104	−	0.999250i	\(-0.512325\pi\)
							0.371068	+	0.928606i	\(0.378992\pi\)
\(23\)	−1.67169	−	2.89545i	−0.348571	−	0.603743i	0.637425	−	0.770513i	\(-0.280000\pi\)
							−0.985996	+	0.166769i	\(0.946666\pi\)
\(29\)	−0.951793	−	2.92932i	−0.176744	−	0.543961i	0.822965	−	0.568092i	\(-0.192318\pi\)
							−0.999709	+	0.0241310i	\(0.992318\pi\)
\(31\)	−6.46796	+	2.87972i	−1.16168	+	0.517213i	−0.894779	−	0.446510i	\(-0.852667\pi\)
							−0.266902	+	0.963724i	\(0.586000\pi\)
\(37\)	−3.01859	+	3.35249i	−0.496254	+	0.551146i	−0.938289	−	0.345852i	\(-0.887590\pi\)
							0.442035	+	0.896998i	\(0.354257\pi\)
\(41\)	0.397318	−	1.22282i	0.0620506	−	0.190972i	−0.915226	−	0.402942i	\(-0.867988\pi\)
							0.977276	+	0.211969i	\(0.0679877\pi\)
\(43\)	−1.59899			−0.243843			−0.121922	−	0.992540i	\(-0.538906\pi\)
							−0.121922	+	0.992540i	\(0.538906\pi\)
\(47\)	1.62042	−	0.344431i	0.236362	−	0.0502404i	−0.0882074	−	0.996102i	\(-0.528114\pi\)
							0.324570	+	0.945862i	\(0.394780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.n.d.81.2		24
7.2	even	3	inner	847.2.n.d.807.2		24
11.2	odd	10		847.2.n.e.753.2		24
11.3	even	5	inner	847.2.n.d.487.2		24
11.4	even	5	inner	847.2.n.d.130.2		24
11.5	even	5		847.2.e.d.606.2		6
11.6	odd	10		77.2.e.b.67.2	yes	6
11.7	odd	10		847.2.n.e.130.2		24
11.8	odd	10		847.2.n.e.487.2		24
11.9	even	5	inner	847.2.n.d.753.2		24
11.10	odd	2		847.2.n.e.81.2		24
33.17	even	10		693.2.i.g.298.2		6
44.39	even	10		1232.2.q.k.529.3		6
77.2	odd	30		847.2.n.e.632.2		24
77.6	even	10		539.2.e.l.67.2		6
77.9	even	15	inner	847.2.n.d.632.2		24
77.16	even	15		847.2.e.d.485.2		6
77.17	even	30		539.2.a.i.1.2		3
77.30	odd	30		847.2.n.e.366.2		24
77.37	even	15	inner	847.2.n.d.9.2		24
77.38	odd	30		5929.2.a.w.1.2		3
77.39	odd	30		539.2.a.h.1.2		3
77.51	odd	30		847.2.n.e.9.2		24
77.58	even	15	inner	847.2.n.d.366.2		24
77.60	even	15		5929.2.a.v.1.2		3
77.61	even	30		539.2.e.l.177.2		6
77.65	odd	6		847.2.n.e.807.2		24
77.72	odd	30		77.2.e.b.23.2	✓	6
231.17	odd	30		4851.2.a.bn.1.2		3
231.116	even	30		4851.2.a.bo.1.2		3
231.149	even	30		693.2.i.g.100.2		6
308.39	even	30		8624.2.a.cl.1.1		3
308.171	odd	30		8624.2.a.ck.1.3		3
308.303	even	30		1232.2.q.k.177.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.2	✓	6	77.72	odd	30
77.2.e.b.67.2	yes	6	11.6	odd	10
539.2.a.h.1.2		3	77.39	odd	30
539.2.a.i.1.2		3	77.17	even	30
539.2.e.l.67.2		6	77.6	even	10
539.2.e.l.177.2		6	77.61	even	30
693.2.i.g.100.2		6	231.149	even	30
693.2.i.g.298.2		6	33.17	even	10
847.2.e.d.485.2		6	77.16	even	15
847.2.e.d.606.2		6	11.5	even	5
847.2.n.d.9.2		24	77.37	even	15	inner
847.2.n.d.81.2		24	1.1	even	1	trivial
847.2.n.d.130.2		24	11.4	even	5	inner
847.2.n.d.366.2		24	77.58	even	15	inner
847.2.n.d.487.2		24	11.3	even	5	inner
847.2.n.d.632.2		24	77.9	even	15	inner
847.2.n.d.753.2		24	11.9	even	5	inner
847.2.n.d.807.2		24	7.2	even	3	inner
847.2.n.e.9.2		24	77.51	odd	30
847.2.n.e.81.2		24	11.10	odd	2
847.2.n.e.130.2		24	11.7	odd	10
847.2.n.e.366.2		24	77.30	odd	30
847.2.n.e.487.2		24	11.8	odd	10
847.2.n.e.632.2		24	77.2	odd	30
847.2.n.e.753.2		24	11.2	odd	10
847.2.n.e.807.2		24	77.65	odd	6
1232.2.q.k.177.3		6	308.303	even	30
1232.2.q.k.529.3		6	44.39	even	10
4851.2.a.bn.1.2		3	231.17	odd	30
4851.2.a.bo.1.2		3	231.116	even	30
5929.2.a.v.1.2		3	77.60	even	15
5929.2.a.w.1.2		3	77.38	odd	30
8624.2.a.ck.1.3		3	308.171	odd	30
8624.2.a.cl.1.1		3	308.39	even	30