Embedded newform 735.2.y.i.128.8

• Label: 735.2.y.i.128.8
• 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			128.8
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.i.557.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.907300 - 0.243110i) q^{2} +(-1.12459 + 1.31730i) q^{3} +(-0.967960 + 0.558852i) q^{4} +(-2.12501 - 0.695932i) q^{5} +(-0.700094 + 1.46859i) q^{6} +(-2.07075 + 2.07075i) q^{8} +(-0.470578 - 2.96286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.907300	−	0.243110i	0.641558	−	0.171905i	0.0766491	−	0.997058i	\(-0.475578\pi\)
							0.564909	+	0.825153i	\(0.308911\pi\)
\(3\)	−1.12459	+	1.31730i	−0.649285	+	0.760546i
							\(5\)	−2.12501	−	0.695932i	−0.950335	−	0.311230i
\(7\)		0			0
							\(11\)	0.630122	−	0.363801i	0.189989	−	0.109690i	−0.401988	−	0.915645i	\(-0.631681\pi\)
0.591977	+	0.805955i	\(0.298347\pi\)
\(13\)	1.44243	+	1.44243i	0.400058	+	0.400058i	0.878253	−	0.478196i	\(-0.158709\pi\)
							−0.478196	+	0.878253i	\(0.658709\pi\)
\(17\)	1.90004	−	7.09105i	0.460828	−	1.71983i	−0.209538	−	0.977801i	\(-0.567196\pi\)
							0.670365	−	0.742031i	\(-0.266137\pi\)
\(19\)	−0.664374	−	0.383576i	−0.152418	−	0.0879985i	0.421851	−	0.906665i	\(-0.361380\pi\)
							−0.574269	+	0.818667i	\(0.694714\pi\)
\(23\)	−0.840245	−	3.13584i	−0.175203	−	0.653867i	−0.996517	−	0.0833906i	\(-0.973425\pi\)
							0.821314	−	0.570477i	\(-0.193242\pi\)
\(29\)	4.07354			0.756437			0.378219	−	0.925716i	\(-0.376537\pi\)
							0.378219	+	0.925716i	\(0.376537\pi\)
\(31\)	0.209930	+	0.363609i	0.0377045	+	0.0653060i	0.884262	−	0.466991i	\(-0.154662\pi\)
							−0.846557	+	0.532297i	\(0.821329\pi\)
\(37\)	−1.63050	−	6.08510i	−0.268052	−	1.00038i	−0.960356	−	0.278778i	\(-0.910071\pi\)
							0.692303	−	0.721607i	\(-0.256596\pi\)
\(41\)		−	4.44452i		−	0.694117i	−0.937843	−	0.347058i	\(-0.887181\pi\)
							0.937843	−	0.347058i	\(-0.112819\pi\)
\(43\)	−5.15881	−	5.15881i	−0.786711	−	0.786711i	0.194243	−	0.980953i	\(-0.437775\pi\)
							−0.980953	+	0.194243i	\(0.937775\pi\)
\(47\)	−6.79316	+	1.82022i	−0.990885	+	0.265507i	−0.717622	−	0.696433i	\(-0.754769\pi\)
							−0.273263	+	0.961939i	\(0.588103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.i.128.8		48
3.2	odd	2	inner	735.2.y.i.128.5		48
5.2	odd	4	inner	735.2.y.i.422.8		48
7.2	even	3		735.2.j.e.638.5		24
7.3	odd	6		105.2.x.a.53.5	yes	48
7.4	even	3	inner	735.2.y.i.263.5		48
7.5	odd	6		735.2.j.g.638.5		24
7.6	odd	2		105.2.x.a.23.8	yes	48
15.2	even	4	inner	735.2.y.i.422.5		48
21.2	odd	6		735.2.j.e.638.8		24
21.5	even	6		735.2.j.g.638.8		24
21.11	odd	6	inner	735.2.y.i.263.8		48
21.17	even	6		105.2.x.a.53.8	yes	48
21.20	even	2		105.2.x.a.23.5	yes	48
35.2	odd	12		735.2.j.e.197.8		24
35.3	even	12		525.2.bf.f.32.8		48
35.12	even	12		735.2.j.g.197.8		24
35.13	even	4		525.2.bf.f.107.5		48
35.17	even	12		105.2.x.a.32.5	yes	48
35.24	odd	6		525.2.bf.f.368.8		48
35.27	even	4		105.2.x.a.2.8	yes	48
35.32	odd	12	inner	735.2.y.i.557.5		48
35.34	odd	2		525.2.bf.f.443.5		48
105.2	even	12		735.2.j.e.197.5		24
105.17	odd	12		105.2.x.a.32.8	yes	48
105.32	even	12	inner	735.2.y.i.557.8		48
105.38	odd	12		525.2.bf.f.32.5		48
105.47	odd	12		735.2.j.g.197.5		24
105.59	even	6		525.2.bf.f.368.5		48
105.62	odd	4		105.2.x.a.2.5	✓	48
105.83	odd	4		525.2.bf.f.107.8		48
105.104	even	2		525.2.bf.f.443.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.5	✓	48	105.62	odd	4
105.2.x.a.2.8	yes	48	35.27	even	4
105.2.x.a.23.5	yes	48	21.20	even	2
105.2.x.a.23.8	yes	48	7.6	odd	2
105.2.x.a.32.5	yes	48	35.17	even	12
105.2.x.a.32.8	yes	48	105.17	odd	12
105.2.x.a.53.5	yes	48	7.3	odd	6
105.2.x.a.53.8	yes	48	21.17	even	6
525.2.bf.f.32.5		48	105.38	odd	12
525.2.bf.f.32.8		48	35.3	even	12
525.2.bf.f.107.5		48	35.13	even	4
525.2.bf.f.107.8		48	105.83	odd	4
525.2.bf.f.368.5		48	105.59	even	6
525.2.bf.f.368.8		48	35.24	odd	6
525.2.bf.f.443.5		48	35.34	odd	2
525.2.bf.f.443.8		48	105.104	even	2
735.2.j.e.197.5		24	105.2	even	12
735.2.j.e.197.8		24	35.2	odd	12
735.2.j.e.638.5		24	7.2	even	3
735.2.j.e.638.8		24	21.2	odd	6
735.2.j.g.197.5		24	105.47	odd	12
735.2.j.g.197.8		24	35.12	even	12
735.2.j.g.638.5		24	7.5	odd	6
735.2.j.g.638.8		24	21.5	even	6
735.2.y.i.128.5		48	3.2	odd	2	inner
735.2.y.i.128.8		48	1.1	even	1	trivial
735.2.y.i.263.5		48	7.4	even	3	inner
735.2.y.i.263.8		48	21.11	odd	6	inner
735.2.y.i.422.5		48	15.2	even	4	inner
735.2.y.i.422.8		48	5.2	odd	4	inner
735.2.y.i.557.5		48	35.32	odd	12	inner
735.2.y.i.557.8		48	105.32	even	12	inner