Embedded newform 735.2.y.i.128.7

• Label: 735.2.y.i.128.7
• Level: $735$
• Weight: $2$
• Character: 735.128
• 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			128.7
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.i.557.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.298314 - 0.0799329i) q^{2} +(1.64547 - 0.540759i) q^{3} +(-1.64945 + 0.952310i) q^{4} +(1.56830 - 1.59387i) q^{5} +(0.447643 - 0.292843i) q^{6} +(-0.852694 + 0.852694i) q^{8} +(2.41516 - 1.77961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.298314	−	0.0799329i	0.210940	−	0.0565211i	−0.151802	−	0.988411i	\(-0.548508\pi\)
							0.362741	+	0.931890i	\(0.381841\pi\)
\(3\)	1.64547	−	0.540759i	0.950014	−	0.312207i
							\(5\)	1.56830	−	1.59387i	0.701367	−	0.712801i
\(7\)		0			0
							\(11\)	0.660315	−	0.381233i	0.199092	−	0.114946i	−0.397140	−	0.917758i	\(-0.629997\pi\)
0.596232	+	0.802812i	\(0.296664\pi\)
\(13\)	2.27077	+	2.27077i	0.629797	+	0.629797i	0.948017	−	0.318220i	\(-0.103085\pi\)
							−0.318220	+	0.948017i	\(0.603085\pi\)
\(17\)	1.25794	−	4.69471i	0.305096	−	1.13864i	−0.627766	−	0.778402i	\(-0.716030\pi\)
							0.932862	−	0.360233i	\(-0.117303\pi\)
\(19\)	−1.41761	−	0.818455i	−0.325221	−	0.187767i	0.328496	−	0.944505i	\(-0.393458\pi\)
							−0.653717	+	0.756739i	\(0.726791\pi\)
\(23\)	1.98015	+	7.39003i	0.412890	+	1.54093i	0.789024	+	0.614363i	\(0.210587\pi\)
							−0.376133	+	0.926566i	\(0.622747\pi\)
\(29\)	4.94251			0.917801			0.458900	−	0.888488i	\(-0.348243\pi\)
							0.458900	+	0.888488i	\(0.348243\pi\)
\(31\)	−2.96413	−	5.13403i	−0.532374	−	0.922099i	−0.999286	−	0.0377949i	\(-0.987967\pi\)
							0.466911	−	0.884304i	\(-0.345367\pi\)
\(37\)	−0.915280	−	3.41587i	−0.150471	−	0.561566i	−0.999451	−	0.0331401i	\(-0.989449\pi\)
							0.848980	−	0.528426i	\(-0.177217\pi\)
\(41\)			4.35963i			0.680860i	0.940270	+	0.340430i	\(0.110573\pi\)
							−0.940270	+	0.340430i	\(0.889427\pi\)
\(43\)	2.69037	+	2.69037i	0.410277	+	0.410277i	0.881835	−	0.471558i	\(-0.156308\pi\)
							−0.471558	+	0.881835i	\(0.656308\pi\)
\(47\)	−4.14148	+	1.10971i	−0.604097	+	0.161867i	−0.547888	−	0.836552i	\(-0.684568\pi\)
							−0.0562089	+	0.998419i	\(0.517901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.i.128.7		48
3.2	odd	2	inner	735.2.y.i.128.6		48
5.2	odd	4	inner	735.2.y.i.422.7		48
7.2	even	3		735.2.j.e.638.6		24
7.3	odd	6		105.2.x.a.53.6	yes	48
7.4	even	3	inner	735.2.y.i.263.6		48
7.5	odd	6		735.2.j.g.638.6		24
7.6	odd	2		105.2.x.a.23.7	yes	48
15.2	even	4	inner	735.2.y.i.422.6		48
21.2	odd	6		735.2.j.e.638.7		24
21.5	even	6		735.2.j.g.638.7		24
21.11	odd	6	inner	735.2.y.i.263.7		48
21.17	even	6		105.2.x.a.53.7	yes	48
21.20	even	2		105.2.x.a.23.6	yes	48
35.2	odd	12		735.2.j.e.197.7		24
35.3	even	12		525.2.bf.f.32.7		48
35.12	even	12		735.2.j.g.197.7		24
35.13	even	4		525.2.bf.f.107.6		48
35.17	even	12		105.2.x.a.32.6	yes	48
35.24	odd	6		525.2.bf.f.368.7		48
35.27	even	4		105.2.x.a.2.7	yes	48
35.32	odd	12	inner	735.2.y.i.557.6		48
35.34	odd	2		525.2.bf.f.443.6		48
105.2	even	12		735.2.j.e.197.6		24
105.17	odd	12		105.2.x.a.32.7	yes	48
105.32	even	12	inner	735.2.y.i.557.7		48
105.38	odd	12		525.2.bf.f.32.6		48
105.47	odd	12		735.2.j.g.197.6		24
105.59	even	6		525.2.bf.f.368.6		48
105.62	odd	4		105.2.x.a.2.6	✓	48
105.83	odd	4		525.2.bf.f.107.7		48
105.104	even	2		525.2.bf.f.443.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.6	✓	48	105.62	odd	4
105.2.x.a.2.7	yes	48	35.27	even	4
105.2.x.a.23.6	yes	48	21.20	even	2
105.2.x.a.23.7	yes	48	7.6	odd	2
105.2.x.a.32.6	yes	48	35.17	even	12
105.2.x.a.32.7	yes	48	105.17	odd	12
105.2.x.a.53.6	yes	48	7.3	odd	6
105.2.x.a.53.7	yes	48	21.17	even	6
525.2.bf.f.32.6		48	105.38	odd	12
525.2.bf.f.32.7		48	35.3	even	12
525.2.bf.f.107.6		48	35.13	even	4
525.2.bf.f.107.7		48	105.83	odd	4
525.2.bf.f.368.6		48	105.59	even	6
525.2.bf.f.368.7		48	35.24	odd	6
525.2.bf.f.443.6		48	35.34	odd	2
525.2.bf.f.443.7		48	105.104	even	2
735.2.j.e.197.6		24	105.2	even	12
735.2.j.e.197.7		24	35.2	odd	12
735.2.j.e.638.6		24	7.2	even	3
735.2.j.e.638.7		24	21.2	odd	6
735.2.j.g.197.6		24	105.47	odd	12
735.2.j.g.197.7		24	35.12	even	12
735.2.j.g.638.6		24	7.5	odd	6
735.2.j.g.638.7		24	21.5	even	6
735.2.y.i.128.6		48	3.2	odd	2	inner
735.2.y.i.128.7		48	1.1	even	1	trivial
735.2.y.i.263.6		48	7.4	even	3	inner
735.2.y.i.263.7		48	21.11	odd	6	inner
735.2.y.i.422.6		48	15.2	even	4	inner
735.2.y.i.422.7		48	5.2	odd	4	inner
735.2.y.i.557.6		48	35.32	odd	12	inner
735.2.y.i.557.7		48	105.32	even	12	inner