Embedded newform 735.2.y.g.557.12

• Label: 735.2.y.g.557.12
• Level: $735$
• Weight: $2$
• Character: 735.557
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.12
Character	\(\chi\)	\(=\)	735.557
Dual form			735.2.y.g.128.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.45834 + 0.658710i) q^{2} +(-1.25586 + 1.19281i) q^{3} +(3.87749 + 2.23867i) q^{4} +(1.98846 + 1.02275i) q^{5} +(-3.87306 + 2.10509i) q^{6} +(4.45829 + 4.45829i) q^{8} +(0.154393 - 2.99602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.45834	+	0.658710i	1.73831	+	0.465779i	0.982070	−	0.188517i	\(-0.0603680\pi\)
							0.756239	+	0.654295i	\(0.227035\pi\)
\(3\)	−1.25586	+	1.19281i	−0.725074	+	0.688671i
							\(5\)	1.98846	+	1.02275i	0.889268	+	0.457386i
\(7\)		0			0
							\(11\)	1.35854	+	0.784351i	0.409614	+	0.236491i	0.690624	−	0.723214i	\(-0.257336\pi\)
−0.281010	+	0.959705i	\(0.590669\pi\)
\(13\)	−2.21881	+	2.21881i	−0.615386	+	0.615386i	−0.944345	−	0.328958i	\(-0.893303\pi\)
							0.328958	+	0.944345i	\(0.393303\pi\)
\(17\)	−1.32034	−	4.92759i	−0.320230	−	1.19512i	−0.919020	−	0.394210i	\(-0.871018\pi\)
							0.598790	−	0.800906i	\(-0.295648\pi\)
\(19\)	1.45527	−	0.840200i	0.333862	−	0.192755i	−0.323693	−	0.946162i	\(-0.604924\pi\)
							0.657554	+	0.753407i	\(0.271591\pi\)
\(23\)	0.364506	−	1.36036i	0.0760049	−	0.283654i	−0.917454	−	0.397841i	\(-0.869760\pi\)
							0.993459	+	0.114187i	\(0.0364263\pi\)
\(29\)	−8.91955			−1.65632			−0.828159	−	0.560493i	\(-0.810612\pi\)
							−0.828159	+	0.560493i	\(0.810612\pi\)
\(31\)	1.37417	−	2.38013i	0.246808	−	0.427484i	−0.715830	−	0.698274i	\(-0.753952\pi\)
							0.962638	+	0.270790i	\(0.0872850\pi\)
\(37\)	0.161183	−	0.601543i	0.0264983	−	0.0988930i	−0.951410	−	0.307926i	\(-0.900365\pi\)
							0.977909	+	0.209033i	\(0.0670317\pi\)
\(41\)		−	6.44292i		−	1.00622i	−0.864224	−	0.503108i	\(-0.832190\pi\)
							0.864224	−	0.503108i	\(-0.167810\pi\)
\(43\)	−5.47734	+	5.47734i	−0.835286	+	0.835286i	−0.988234	−	0.152948i	\(-0.951123\pi\)
							0.152948	+	0.988234i	\(0.451123\pi\)
\(47\)	5.04552	+	1.35194i	0.735965	+	0.197201i	0.607284	−	0.794485i	\(-0.292259\pi\)
							0.128681	+	0.991686i	\(0.458926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.g.557.12		48
3.2	odd	2	inner	735.2.y.g.557.1		48
5.3	odd	4	inner	735.2.y.g.263.12		48
7.2	even	3	inner	735.2.y.g.422.1		48
7.3	odd	6		105.2.j.a.92.1	yes	24
7.4	even	3		735.2.j.h.197.1		24
7.5	odd	6		735.2.y.j.422.1		48
7.6	odd	2		735.2.y.j.557.12		48
15.8	even	4	inner	735.2.y.g.263.1		48
21.2	odd	6	inner	735.2.y.g.422.12		48
21.5	even	6		735.2.y.j.422.12		48
21.11	odd	6		735.2.j.h.197.12		24
21.17	even	6		105.2.j.a.92.12	yes	24
21.20	even	2		735.2.y.j.557.1		48
35.3	even	12		105.2.j.a.8.12	yes	24
35.13	even	4		735.2.y.j.263.12		48
35.17	even	12		525.2.j.b.218.1		24
35.18	odd	12		735.2.j.h.638.12		24
35.23	odd	12	inner	735.2.y.g.128.1		48
35.24	odd	6		525.2.j.b.407.12		24
35.33	even	12		735.2.y.j.128.1		48
105.17	odd	12		525.2.j.b.218.12		24
105.23	even	12	inner	735.2.y.g.128.12		48
105.38	odd	12		105.2.j.a.8.1	✓	24
105.53	even	12		735.2.j.h.638.1		24
105.59	even	6		525.2.j.b.407.1		24
105.68	odd	12		735.2.y.j.128.12		48
105.83	odd	4		735.2.y.j.263.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.1	✓	24	105.38	odd	12
105.2.j.a.8.12	yes	24	35.3	even	12
105.2.j.a.92.1	yes	24	7.3	odd	6
105.2.j.a.92.12	yes	24	21.17	even	6
525.2.j.b.218.1		24	35.17	even	12
525.2.j.b.218.12		24	105.17	odd	12
525.2.j.b.407.1		24	105.59	even	6
525.2.j.b.407.12		24	35.24	odd	6
735.2.j.h.197.1		24	7.4	even	3
735.2.j.h.197.12		24	21.11	odd	6
735.2.j.h.638.1		24	105.53	even	12
735.2.j.h.638.12		24	35.18	odd	12
735.2.y.g.128.1		48	35.23	odd	12	inner
735.2.y.g.128.12		48	105.23	even	12	inner
735.2.y.g.263.1		48	15.8	even	4	inner
735.2.y.g.263.12		48	5.3	odd	4	inner
735.2.y.g.422.1		48	7.2	even	3	inner
735.2.y.g.422.12		48	21.2	odd	6	inner
735.2.y.g.557.1		48	3.2	odd	2	inner
735.2.y.g.557.12		48	1.1	even	1	trivial
735.2.y.j.128.1		48	35.33	even	12
735.2.y.j.128.12		48	105.68	odd	12
735.2.y.j.263.1		48	105.83	odd	4
735.2.y.j.263.12		48	35.13	even	4
735.2.y.j.422.1		48	7.5	odd	6
735.2.y.j.422.12		48	21.5	even	6
735.2.y.j.557.1		48	21.20	even	2
735.2.y.j.557.12		48	7.6	odd	2