Embedded newform 105.2.x.a.53.2

• Label: 105.2.x.a.53.2
• Level: $105$
• Weight: $2$
• Character: 105.53
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			53.2
Character	\(\chi\)	\(=\)	105.53
Dual form			105.2.x.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.582118 + 2.17249i) q^{2} +(1.60750 - 0.644934i) q^{3} +(-2.64881 - 1.52929i) q^{4} +(2.21039 - 0.337883i) q^{5} +(0.465359 + 3.86771i) q^{6} +(-1.15099 + 2.38227i) q^{7} +(1.68355 - 1.68355i) q^{8} +(2.16812 - 2.07346i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.582118	+	2.17249i	−0.411619	+	1.53618i	0.379893	+	0.925031i	\(0.375961\pi\)
							−0.791512	+	0.611154i	\(0.790706\pi\)
\(3\)	1.60750	−	0.644934i	0.928091	−	0.372353i
							\(5\)	2.21039	−	0.337883i	0.988518	−	0.151106i
\(7\)	−1.15099	+	2.38227i	−0.435034	+	0.900414i
							\(11\)	−3.88729	−	2.24433i	−1.17206	−	0.676690i	−0.217896	−	0.975972i	\(-0.569919\pi\)
−0.954165	+	0.299282i	\(0.903253\pi\)
\(13\)	−1.08424	−	1.08424i	−0.300715	−	0.300715i	0.540578	−	0.841294i	\(-0.318205\pi\)
							−0.841294	+	0.540578i	\(0.818205\pi\)
\(17\)	2.04863	−	0.548929i	0.496866	−	0.133135i	−0.00167924	−	0.999999i	\(-0.500535\pi\)
							0.498545	+	0.866864i	\(0.333868\pi\)
\(19\)	−3.66075	+	2.11354i	−0.839834	+	0.484878i	−0.857208	−	0.514971i	\(-0.827803\pi\)
							0.0173737	+	0.999849i	\(0.494469\pi\)
\(23\)	3.13628	+	0.840363i	0.653959	+	0.175228i	0.570518	−	0.821285i	\(-0.306742\pi\)
							0.0834407	+	0.996513i	\(0.473409\pi\)
\(29\)	−1.69118			−0.314045			−0.157023	−	0.987595i	\(-0.550190\pi\)
							−0.157023	+	0.987595i	\(0.550190\pi\)
\(31\)	0.530077	−	0.918121i	0.0952047	−	0.164899i	−0.814489	−	0.580179i	\(-0.802983\pi\)
							0.909694	+	0.415279i	\(0.136316\pi\)
\(37\)	−5.75771	−	1.54277i	−0.946562	−	0.253631i	−0.247659	−	0.968847i	\(-0.579661\pi\)
							−0.698903	+	0.715217i	\(0.746328\pi\)
\(41\)		−	5.84230i		−	0.912414i	−0.889874	−	0.456207i	\(-0.849208\pi\)
							0.889874	−	0.456207i	\(-0.150792\pi\)
\(43\)	2.00369	+	2.00369i	0.305559	+	0.305559i	0.843184	−	0.537625i	\(-0.180678\pi\)
							−0.537625	+	0.843184i	\(0.680678\pi\)
\(47\)	−1.36662	+	5.10030i	−0.199342	+	0.743956i	0.791758	+	0.610835i	\(0.209166\pi\)
							−0.991100	+	0.133120i	\(0.957500\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.x.a.53.2	yes	48
3.2	odd	2	inner	105.2.x.a.53.11	yes	48
5.2	odd	4	inner	105.2.x.a.32.2	yes	48
5.3	odd	4		525.2.bf.f.32.11		48
5.4	even	2		525.2.bf.f.368.11		48
7.2	even	3	inner	105.2.x.a.23.11	yes	48
7.3	odd	6		735.2.j.e.638.2		24
7.4	even	3		735.2.j.g.638.2		24
7.5	odd	6		735.2.y.i.128.11		48
7.6	odd	2		735.2.y.i.263.2		48
15.2	even	4	inner	105.2.x.a.32.11	yes	48
15.8	even	4		525.2.bf.f.32.2		48
15.14	odd	2		525.2.bf.f.368.2		48
21.2	odd	6	inner	105.2.x.a.23.2	yes	48
21.5	even	6		735.2.y.i.128.2		48
21.11	odd	6		735.2.j.g.638.11		24
21.17	even	6		735.2.j.e.638.11		24
21.20	even	2		735.2.y.i.263.11		48
35.2	odd	12	inner	105.2.x.a.2.11	yes	48
35.9	even	6		525.2.bf.f.443.2		48
35.12	even	12		735.2.y.i.422.11		48
35.17	even	12		735.2.j.e.197.11		24
35.23	odd	12		525.2.bf.f.107.2		48
35.27	even	4		735.2.y.i.557.2		48
35.32	odd	12		735.2.j.g.197.11		24
105.2	even	12	inner	105.2.x.a.2.2	✓	48
105.17	odd	12		735.2.j.e.197.2		24
105.23	even	12		525.2.bf.f.107.11		48
105.32	even	12		735.2.j.g.197.2		24
105.44	odd	6		525.2.bf.f.443.11		48
105.47	odd	12		735.2.y.i.422.2		48
105.62	odd	4		735.2.y.i.557.11		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.2	✓	48	105.2	even	12	inner
105.2.x.a.2.11	yes	48	35.2	odd	12	inner
105.2.x.a.23.2	yes	48	21.2	odd	6	inner
105.2.x.a.23.11	yes	48	7.2	even	3	inner
105.2.x.a.32.2	yes	48	5.2	odd	4	inner
105.2.x.a.32.11	yes	48	15.2	even	4	inner
105.2.x.a.53.2	yes	48	1.1	even	1	trivial
105.2.x.a.53.11	yes	48	3.2	odd	2	inner
525.2.bf.f.32.2		48	15.8	even	4
525.2.bf.f.32.11		48	5.3	odd	4
525.2.bf.f.107.2		48	35.23	odd	12
525.2.bf.f.107.11		48	105.23	even	12
525.2.bf.f.368.2		48	15.14	odd	2
525.2.bf.f.368.11		48	5.4	even	2
525.2.bf.f.443.2		48	35.9	even	6
525.2.bf.f.443.11		48	105.44	odd	6
735.2.j.e.197.2		24	105.17	odd	12
735.2.j.e.197.11		24	35.17	even	12
735.2.j.e.638.2		24	7.3	odd	6
735.2.j.e.638.11		24	21.17	even	6
735.2.j.g.197.2		24	105.32	even	12
735.2.j.g.197.11		24	35.32	odd	12
735.2.j.g.638.2		24	7.4	even	3
735.2.j.g.638.11		24	21.11	odd	6
735.2.y.i.128.2		48	21.5	even	6
735.2.y.i.128.11		48	7.5	odd	6
735.2.y.i.263.2		48	7.6	odd	2
735.2.y.i.263.11		48	21.20	even	2
735.2.y.i.422.2		48	105.47	odd	12
735.2.y.i.422.11		48	35.12	even	12
735.2.y.i.557.2		48	35.27	even	4
735.2.y.i.557.11		48	105.62	odd	4