Embedded newform 1216.2.i.d.961.1

• Label: 1216.2.i.d.961.1
• Level: $1216$
• Weight: $2$
• Character: 1216.961
• Analytic conductor: $9.710$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.70980888579\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1216.961
Dual form			1216.2.i.d.577.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +4.00000 q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(705\)	\(837\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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			0
							\(3\)	−0.500000	+	0.866025i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	\(-0.926548\pi\)
0.684819	+	0.728714i	\(0.259881\pi\)
\(5\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)	4.00000			1.51186			0.755929	−	0.654654i	\(-0.227186\pi\)
							0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	3.00000			0.904534			0.452267	−	0.891883i	\(-0.350615\pi\)
							0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	\(-0.0772105\pi\)
							−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	\(-0.573966\pi\)
							0.957892	−	0.287129i	\(-0.0927008\pi\)
\(19\)	−3.50000	−	2.59808i	−0.802955	−	0.596040i
							\(23\)	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	\(-0.951544\pi\)
0.362892	−	0.931831i	\(-0.381789\pi\)
\(29\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(31\)	−2.00000			−0.359211			−0.179605	−	0.983739i	\(-0.557482\pi\)
							−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	10.0000			1.64399			0.821995	−	0.569495i	\(-0.192861\pi\)
							0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	\(0.414726\pi\)
							−0.967486	+	0.252924i	\(0.918608\pi\)
\(43\)	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	\(-0.734678\pi\)
							0.977261	+	0.212041i	\(0.0680112\pi\)
\(47\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.i.d.961.1		2
4.3	odd	2		1216.2.i.h.961.1		2
8.3	odd	2		38.2.c.a.11.1	yes	2
8.5	even	2		304.2.i.c.49.1		2
19.7	even	3	inner	1216.2.i.d.577.1		2
24.5	odd	2		2736.2.s.m.1873.1		2
24.11	even	2		342.2.g.b.163.1		2
40.3	even	4		950.2.j.e.49.1		4
40.19	odd	2		950.2.e.d.201.1		2
40.27	even	4		950.2.j.e.49.2		4
76.7	odd	6		1216.2.i.h.577.1		2
152.3	even	18		722.2.e.i.99.1		6
152.11	odd	6		722.2.a.c.1.1		1
152.27	even	6		722.2.a.d.1.1		1
152.35	odd	18		722.2.e.j.99.1		6
152.43	odd	18		722.2.e.j.389.1		6
152.45	even	6		304.2.i.c.273.1		2
152.51	even	18		722.2.e.i.415.1		6
152.59	even	18		722.2.e.i.595.1		6
152.67	even	18		722.2.e.i.245.1		6
152.75	even	2		722.2.c.b.429.1		2
152.83	odd	6		38.2.c.a.7.1	✓	2
152.91	even	18		722.2.e.i.423.1		6
152.99	odd	18		722.2.e.j.423.1		6
152.107	even	6		722.2.c.b.653.1		2
152.123	odd	18		722.2.e.j.245.1		6
152.125	even	6		5776.2.a.g.1.1		1
152.131	odd	18		722.2.e.j.595.1		6
152.139	odd	18		722.2.e.j.415.1		6
152.141	odd	6		5776.2.a.n.1.1		1
152.147	even	18		722.2.e.i.389.1		6
456.11	even	6		6498.2.a.s.1.1		1
456.83	even	6		342.2.g.b.235.1		2
456.179	odd	6		6498.2.a.e.1.1		1
456.197	odd	6		2736.2.s.m.577.1		2
760.83	even	12		950.2.j.e.349.2		4
760.387	even	12		950.2.j.e.349.1		4
760.539	odd	6		950.2.e.d.501.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.a.7.1	✓	2	152.83	odd	6
38.2.c.a.11.1	yes	2	8.3	odd	2
304.2.i.c.49.1		2	8.5	even	2
304.2.i.c.273.1		2	152.45	even	6
342.2.g.b.163.1		2	24.11	even	2
342.2.g.b.235.1		2	456.83	even	6
722.2.a.c.1.1		1	152.11	odd	6
722.2.a.d.1.1		1	152.27	even	6
722.2.c.b.429.1		2	152.75	even	2
722.2.c.b.653.1		2	152.107	even	6
722.2.e.i.99.1		6	152.3	even	18
722.2.e.i.245.1		6	152.67	even	18
722.2.e.i.389.1		6	152.147	even	18
722.2.e.i.415.1		6	152.51	even	18
722.2.e.i.423.1		6	152.91	even	18
722.2.e.i.595.1		6	152.59	even	18
722.2.e.j.99.1		6	152.35	odd	18
722.2.e.j.245.1		6	152.123	odd	18
722.2.e.j.389.1		6	152.43	odd	18
722.2.e.j.415.1		6	152.139	odd	18
722.2.e.j.423.1		6	152.99	odd	18
722.2.e.j.595.1		6	152.131	odd	18
950.2.e.d.201.1		2	40.19	odd	2
950.2.e.d.501.1		2	760.539	odd	6
950.2.j.e.49.1		4	40.3	even	4
950.2.j.e.49.2		4	40.27	even	4
950.2.j.e.349.1		4	760.387	even	12
950.2.j.e.349.2		4	760.83	even	12
1216.2.i.d.577.1		2	19.7	even	3	inner
1216.2.i.d.961.1		2	1.1	even	1	trivial
1216.2.i.h.577.1		2	76.7	odd	6
1216.2.i.h.961.1		2	4.3	odd	2
2736.2.s.m.577.1		2	456.197	odd	6
2736.2.s.m.1873.1		2	24.5	odd	2
5776.2.a.g.1.1		1	152.125	even	6
5776.2.a.n.1.1		1	152.141	odd	6
6498.2.a.e.1.1		1	456.179	odd	6
6498.2.a.s.1.1		1	456.11	even	6