Embedded newform 735.2.y.i.128.10

• Label: 735.2.y.i.128.10
• 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			128.10
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.i.557.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46015 - 0.391246i) q^{2} +(0.852980 + 1.50746i) q^{3} +(0.246919 - 0.142558i) q^{4} +(-0.207883 + 2.22638i) q^{5} +(1.83527 + 1.86739i) q^{6} +(-1.83305 + 1.83305i) q^{8} +(-1.54485 + 2.57166i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46015	−	0.391246i	1.03248	−	0.276653i	0.297488	−	0.954726i	\(-0.403851\pi\)
							0.734995	+	0.678072i	\(0.237184\pi\)
\(3\)	0.852980	+	1.50746i	0.492468	+	0.870330i
							\(5\)	−0.207883	+	2.22638i	−0.0929679	+	0.995669i
\(7\)		0			0
							\(11\)	0.791646	−	0.457057i	0.238690	−	0.137808i	−0.375884	−	0.926667i	\(-0.622661\pi\)
0.614575	+	0.788859i	\(0.289328\pi\)
\(13\)	−3.07974	−	3.07974i	−0.854166	−	0.854166i	0.136477	−	0.990643i	\(-0.456422\pi\)
							−0.990643	+	0.136477i	\(0.956422\pi\)
\(17\)	0.311437	−	1.16230i	0.0755345	−	0.281899i	−0.917819	−	0.396998i	\(-0.870052\pi\)
							0.993354	+	0.115099i	\(0.0367187\pi\)
\(19\)	5.95337	+	3.43718i	1.36580	+	0.788543i	0.990388	−	0.138316i	\(-0.0441690\pi\)
							0.375409	+	0.926859i	\(0.377502\pi\)
\(23\)	0.505926	+	1.88814i	0.105493	+	0.393705i	0.998401	−	0.0565348i	\(-0.0180052\pi\)
							−0.892908	+	0.450240i	\(0.851339\pi\)
\(29\)	2.72261			0.505576			0.252788	−	0.967522i	\(-0.418652\pi\)
							0.252788	+	0.967522i	\(0.418652\pi\)
\(31\)	2.31688	+	4.01295i	0.416123	+	0.720747i	0.995546	−	0.0942806i	\(-0.0300551\pi\)
							−0.579422	+	0.815028i	\(0.696722\pi\)
\(37\)	0.207656	+	0.774982i	0.0341384	+	0.127406i	0.980892	−	0.194552i	\(-0.0623255\pi\)
							−0.946754	+	0.321959i	\(0.895659\pi\)
\(41\)			0.922837i			0.144123i	0.997400	+	0.0720615i	\(0.0229578\pi\)
							−0.997400	+	0.0720615i	\(0.977042\pi\)
\(43\)	−4.80893	−	4.80893i	−0.733355	−	0.733355i	0.237928	−	0.971283i	\(-0.423532\pi\)
							−0.971283	+	0.237928i	\(0.923532\pi\)
\(47\)	10.1240	−	2.71272i	1.47674	−	0.395691i	0.571503	−	0.820600i	\(-0.306360\pi\)
							0.905236	+	0.424909i	\(0.139694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.i.128.10		48
3.2	odd	2	inner	735.2.y.i.128.3		48
5.2	odd	4	inner	735.2.y.i.422.10		48
7.2	even	3		735.2.j.e.638.3		24
7.3	odd	6		105.2.x.a.53.3	yes	48
7.4	even	3	inner	735.2.y.i.263.3		48
7.5	odd	6		735.2.j.g.638.3		24
7.6	odd	2		105.2.x.a.23.10	yes	48
15.2	even	4	inner	735.2.y.i.422.3		48
21.2	odd	6		735.2.j.e.638.10		24
21.5	even	6		735.2.j.g.638.10		24
21.11	odd	6	inner	735.2.y.i.263.10		48
21.17	even	6		105.2.x.a.53.10	yes	48
21.20	even	2		105.2.x.a.23.3	yes	48
35.2	odd	12		735.2.j.e.197.10		24
35.3	even	12		525.2.bf.f.32.10		48
35.12	even	12		735.2.j.g.197.10		24
35.13	even	4		525.2.bf.f.107.3		48
35.17	even	12		105.2.x.a.32.3	yes	48
35.24	odd	6		525.2.bf.f.368.10		48
35.27	even	4		105.2.x.a.2.10	yes	48
35.32	odd	12	inner	735.2.y.i.557.3		48
35.34	odd	2		525.2.bf.f.443.3		48
105.2	even	12		735.2.j.e.197.3		24
105.17	odd	12		105.2.x.a.32.10	yes	48
105.32	even	12	inner	735.2.y.i.557.10		48
105.38	odd	12		525.2.bf.f.32.3		48
105.47	odd	12		735.2.j.g.197.3		24
105.59	even	6		525.2.bf.f.368.3		48
105.62	odd	4		105.2.x.a.2.3	✓	48
105.83	odd	4		525.2.bf.f.107.10		48
105.104	even	2		525.2.bf.f.443.10		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.3	✓	48	105.62	odd	4
105.2.x.a.2.10	yes	48	35.27	even	4
105.2.x.a.23.3	yes	48	21.20	even	2
105.2.x.a.23.10	yes	48	7.6	odd	2
105.2.x.a.32.3	yes	48	35.17	even	12
105.2.x.a.32.10	yes	48	105.17	odd	12
105.2.x.a.53.3	yes	48	7.3	odd	6
105.2.x.a.53.10	yes	48	21.17	even	6
525.2.bf.f.32.3		48	105.38	odd	12
525.2.bf.f.32.10		48	35.3	even	12
525.2.bf.f.107.3		48	35.13	even	4
525.2.bf.f.107.10		48	105.83	odd	4
525.2.bf.f.368.3		48	105.59	even	6
525.2.bf.f.368.10		48	35.24	odd	6
525.2.bf.f.443.3		48	35.34	odd	2
525.2.bf.f.443.10		48	105.104	even	2
735.2.j.e.197.3		24	105.2	even	12
735.2.j.e.197.10		24	35.2	odd	12
735.2.j.e.638.3		24	7.2	even	3
735.2.j.e.638.10		24	21.2	odd	6
735.2.j.g.197.3		24	105.47	odd	12
735.2.j.g.197.10		24	35.12	even	12
735.2.j.g.638.3		24	7.5	odd	6
735.2.j.g.638.10		24	21.5	even	6
735.2.y.i.128.3		48	3.2	odd	2	inner
735.2.y.i.128.10		48	1.1	even	1	trivial
735.2.y.i.263.3		48	7.4	even	3	inner
735.2.y.i.263.10		48	21.11	odd	6	inner
735.2.y.i.422.3		48	15.2	even	4	inner
735.2.y.i.422.10		48	5.2	odd	4	inner
735.2.y.i.557.3		48	35.32	odd	12	inner
735.2.y.i.557.10		48	105.32	even	12	inner