Embedded newform 847.2.n.e.753.2

• Label: 847.2.n.e.753.2
• Level: $847$
• Weight: $2$
• Character: 847.753
• 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			753.2
Character	\(\chi\)	\(=\)	847.753
Dual form			847.2.n.e.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.439365 - 0.487964i) q^{2} +(1.74691 - 0.777774i) q^{3} +(0.163989 + 1.56026i) q^{4} +(3.49086 - 0.742006i) q^{5} +(0.388004 - 1.19415i) q^{6} +(-0.295442 + 2.62920i) q^{7} +(1.89583 + 1.37740i) q^{8} +(0.439365 - 0.487964i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 18 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.439365	−	0.487964i	0.310678	−	0.345043i	−0.567503	−	0.823371i	\(-0.692091\pi\)
							0.878181	+	0.478329i	\(0.158757\pi\)
\(3\)	1.74691	−	0.777774i	1.00858	−	0.449048i	0.165138	−	0.986271i	\(-0.447193\pi\)
							0.843440	+	0.537223i	\(0.180527\pi\)
\(5\)	3.49086	−	0.742006i	1.56116	−	0.331835i	0.655286	−	0.755381i	\(-0.272548\pi\)
							0.905875	+	0.423546i	\(0.139215\pi\)
\(7\)	−0.295442	+	2.62920i	−0.111666	+	0.993746i
							\(11\)		0			0
\(13\)	−1.82698	−	5.62286i	−0.506713	−	1.55950i								−0.797872	−	0.602827i	\(-0.794041\pi\)
							0.291159	−	0.956675i	\(-0.405959\pi\)
\(17\)	−1.10850	−	1.23111i	−0.268850	−	0.298588i	0.593569	−	0.804783i	\(-0.297718\pi\)
							−0.862419	+	0.506195i	\(0.831052\pi\)
\(19\)	−0.154815	+	1.47297i	−0.0355170	+	0.337921i	0.962306	+	0.271970i	\(0.0876750\pi\)
							−0.997823	+	0.0659518i	\(0.978992\pi\)
\(23\)	−1.67169	−	2.89545i	−0.348571	−	0.603743i	0.637425	−	0.770513i	\(-0.280000\pi\)
							−0.985996	+	0.166769i	\(0.946666\pi\)
\(29\)	−2.49183	+	1.81042i	−0.462721	+	0.336186i	−0.794598	−	0.607136i	\(-0.792318\pi\)
							0.331877	+	0.943323i	\(0.392318\pi\)
\(31\)	6.92535	+	1.47203i	1.24383	+	0.264384i	0.782391	−	0.622788i	\(-0.214000\pi\)
							0.461439	+	0.887172i	\(0.347333\pi\)
\(37\)	−4.12120	−	1.83488i	−0.677521	−	0.301652i	0.0389776	−	0.999240i	\(-0.487590\pi\)
							−0.716499	+	0.697588i	\(0.754257\pi\)
\(41\)	1.04019	+	0.755743i	0.162451	+	0.118027i	0.666041	−	0.745916i	\(-0.267988\pi\)
							−0.503590	+	0.863943i	\(0.667988\pi\)
\(43\)	1.59899			0.243843			0.121922	−	0.992540i	\(-0.461094\pi\)
							0.121922	+	0.992540i	\(0.461094\pi\)
\(47\)	0.173164	−	1.64755i	0.0252586	−	0.240319i	−0.974607	−	0.223922i	\(-0.928114\pi\)
							0.999866	−	0.0163969i	\(-0.00521952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.n.e.753.2		24
7.2	even	3	inner	847.2.n.e.632.2		24
11.2	odd	10		847.2.n.d.130.2		24
11.3	even	5		77.2.e.b.67.2	yes	6
11.4	even	5	inner	847.2.n.e.487.2		24
11.5	even	5	inner	847.2.n.e.81.2		24
11.6	odd	10		847.2.n.d.81.2		24
11.7	odd	10		847.2.n.d.487.2		24
11.8	odd	10		847.2.e.d.606.2		6
11.9	even	5	inner	847.2.n.e.130.2		24
11.10	odd	2		847.2.n.d.753.2		24
33.14	odd	10		693.2.i.g.298.2		6
44.3	odd	10		1232.2.q.k.529.3		6
77.2	odd	30		847.2.n.d.9.2		24
77.3	odd	30		539.2.a.i.1.2		3
77.9	even	15	inner	847.2.n.e.9.2		24
77.16	even	15	inner	847.2.n.e.807.2		24
77.25	even	15		539.2.a.h.1.2		3
77.30	odd	30		847.2.e.d.485.2		6
77.37	even	15	inner	847.2.n.e.366.2		24
77.47	odd	30		539.2.e.l.177.2		6
77.51	odd	30		847.2.n.d.366.2		24
77.52	even	30		5929.2.a.w.1.2		3
77.58	even	15		77.2.e.b.23.2	✓	6
77.65	odd	6		847.2.n.d.632.2		24
77.69	odd	10		539.2.e.l.67.2		6
77.72	odd	30		847.2.n.d.807.2		24
77.74	odd	30		5929.2.a.v.1.2		3
231.80	even	30		4851.2.a.bn.1.2		3
231.179	odd	30		4851.2.a.bo.1.2		3
231.212	odd	30		693.2.i.g.100.2		6
308.3	even	30		8624.2.a.ck.1.3		3
308.135	odd	30		1232.2.q.k.177.3		6
308.179	odd	30		8624.2.a.cl.1.1		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.2	✓	6	77.58	even	15
77.2.e.b.67.2	yes	6	11.3	even	5
539.2.a.h.1.2		3	77.25	even	15
539.2.a.i.1.2		3	77.3	odd	30
539.2.e.l.67.2		6	77.69	odd	10
539.2.e.l.177.2		6	77.47	odd	30
693.2.i.g.100.2		6	231.212	odd	30
693.2.i.g.298.2		6	33.14	odd	10
847.2.e.d.485.2		6	77.30	odd	30
847.2.e.d.606.2		6	11.8	odd	10
847.2.n.d.9.2		24	77.2	odd	30
847.2.n.d.81.2		24	11.6	odd	10
847.2.n.d.130.2		24	11.2	odd	10
847.2.n.d.366.2		24	77.51	odd	30
847.2.n.d.487.2		24	11.7	odd	10
847.2.n.d.632.2		24	77.65	odd	6
847.2.n.d.753.2		24	11.10	odd	2
847.2.n.d.807.2		24	77.72	odd	30
847.2.n.e.9.2		24	77.9	even	15	inner
847.2.n.e.81.2		24	11.5	even	5	inner
847.2.n.e.130.2		24	11.9	even	5	inner
847.2.n.e.366.2		24	77.37	even	15	inner
847.2.n.e.487.2		24	11.4	even	5	inner
847.2.n.e.632.2		24	7.2	even	3	inner
847.2.n.e.753.2		24	1.1	even	1	trivial
847.2.n.e.807.2		24	77.16	even	15	inner
1232.2.q.k.177.3		6	308.135	odd	30
1232.2.q.k.529.3		6	44.3	odd	10
4851.2.a.bn.1.2		3	231.80	even	30
4851.2.a.bo.1.2		3	231.179	odd	30
5929.2.a.v.1.2		3	77.74	odd	30
5929.2.a.w.1.2		3	77.52	even	30
8624.2.a.ck.1.3		3	308.3	even	30
8624.2.a.cl.1.1		3	308.179	odd	30