Embedded newform 735.2.y.g.422.1

• Label: 735.2.y.g.422.1
• Level: $735$
• Weight: $2$
• Character: 735.422
• 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			422.1
Character	\(\chi\)	\(=\)	735.422
Dual form			735.2.y.g.263.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.658710 - 2.45834i) q^{2} +(-0.405074 - 1.68402i) q^{3} +(-3.87749 + 2.23867i) q^{4} +(-0.108509 - 2.23343i) q^{5} +(-3.87306 + 2.10509i) q^{6} +(4.45829 + 4.45829i) q^{8} +(-2.67183 + 1.36430i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.658710	−	2.45834i	−0.465779	−	1.73831i	−0.654295	−	0.756239i	\(-0.727035\pi\)
							0.188517	−	0.982070i	\(-0.439632\pi\)
\(3\)	−0.405074	−	1.68402i	−0.233870	−	0.972268i
							\(5\)	−0.108509	−	2.23343i	−0.0485266	−	0.998822i
\(7\)		0			0
							\(11\)	−1.35854	+	0.784351i	−0.409614	+	0.236491i	−0.690624	−	0.723214i	\(-0.742664\pi\)
0.281010	+	0.959705i	\(0.409331\pi\)
\(13\)	−2.21881	+	2.21881i	−0.615386	+	0.615386i	−0.944345	−	0.328958i	\(-0.893303\pi\)
							0.328958	+	0.944345i	\(0.393303\pi\)
\(17\)	4.92759	+	1.32034i	1.19512	+	0.320230i	0.800906	−	0.598790i	\(-0.204352\pi\)
							0.394210	+	0.919020i	\(0.371018\pi\)
\(19\)	−1.45527	−	0.840200i	−0.333862	−	0.192755i	0.323693	−	0.946162i	\(-0.395076\pi\)
							−0.657554	+	0.753407i	\(0.728409\pi\)
\(23\)	−1.36036	+	0.364506i	−0.283654	+	0.0760049i	−0.397841	−	0.917454i	\(-0.630240\pi\)
							0.114187	+	0.993459i	\(0.463574\pi\)
\(29\)	−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	0.962638	−	0.270790i	\(-0.0872850\pi\)
							−0.715830	+	0.698274i	\(0.753952\pi\)
\(37\)	−0.601543	+	0.161183i	−0.0988930	+	0.0264983i	−0.307926	−	0.951410i	\(-0.599635\pi\)
							0.209033	+	0.977909i	\(0.432968\pi\)
\(41\)		−	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	\(0.451123\pi\)
\(47\)	−1.35194	−	5.04552i	−0.197201	−	0.735965i	−0.991686	−	0.128681i	\(-0.958926\pi\)
							0.794485	−	0.607284i	\(-0.207741\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.1	✓	24	105.68	odd	12
105.2.j.a.8.12	yes	24	35.33	even	12
105.2.j.a.92.1	yes	24	7.5	odd	6
105.2.j.a.92.12	yes	24	21.5	even	6
525.2.j.b.218.1		24	35.12	even	12
525.2.j.b.218.12		24	105.47	odd	12
525.2.j.b.407.1		24	105.89	even	6
525.2.j.b.407.12		24	35.19	odd	6
735.2.j.h.197.1		24	7.2	even	3
735.2.j.h.197.12		24	21.2	odd	6
735.2.j.h.638.1		24	105.23	even	12
735.2.j.h.638.12		24	35.23	odd	12
735.2.y.g.128.1		48	5.3	odd	4	inner
735.2.y.g.128.12		48	15.8	even	4	inner
735.2.y.g.263.1		48	105.53	even	12	inner
735.2.y.g.263.12		48	35.18	odd	12	inner
735.2.y.g.422.1		48	1.1	even	1	trivial
735.2.y.g.422.12		48	3.2	odd	2	inner
735.2.y.g.557.1		48	21.11	odd	6	inner
735.2.y.g.557.12		48	7.4	even	3	inner
735.2.y.j.128.1		48	35.13	even	4
735.2.y.j.128.12		48	105.83	odd	4
735.2.y.j.263.1		48	105.38	odd	12
735.2.y.j.263.12		48	35.3	even	12
735.2.y.j.422.1		48	7.6	odd	2
735.2.y.j.422.12		48	21.20	even	2
735.2.y.j.557.1		48	21.17	even	6
735.2.y.j.557.12		48	7.3	odd	6