Embedded newform 525.2.bf.f.32.2

• Label: 525.2.bf.f.32.2
• Level: $525$
• Weight: $2$
• Character: 525.32
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
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			32.2
Character	\(\chi\)	\(=\)	525.32
Dual form			525.2.bf.f.443.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.17249 - 0.582118i) q^{2} +(0.245221 + 1.71460i) q^{3} +(2.64881 + 1.52929i) q^{4} +(0.465359 - 3.86771i) q^{6} +(2.38227 + 1.15099i) q^{7} +(-1.68355 - 1.68355i) q^{8} +(-2.87973 + 0.840915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} - 24 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.17249	−	0.582118i	−1.53618	−	0.411619i	−0.611154	−	0.791512i	\(-0.709294\pi\)
							−0.925031	+	0.379893i	\(0.875961\pi\)
\(3\)	0.245221	+	1.71460i	0.141579	+	0.989927i
							\(5\)		0			0
\(7\)	2.38227	+	1.15099i	0.900414	+	0.435034i
							\(11\)	3.88729	+	2.24433i	1.17206	+	0.676690i	0.954165	−	0.299282i	\(-0.0967473\pi\)
0.217896	+	0.975972i	\(0.430081\pi\)
\(13\)	1.08424	−	1.08424i	0.300715	−	0.300715i	−0.540578	−	0.841294i	\(-0.681795\pi\)
							0.841294	+	0.540578i	\(0.181795\pi\)
\(17\)	0.548929	+	2.04863i	0.133135	+	0.496866i	0.999999	−	0.00167924i	\(-0.000534519\pi\)
							−0.866864	+	0.498545i	\(0.833868\pi\)
\(19\)	3.66075	−	2.11354i	0.839834	−	0.484878i	−0.0173737	−	0.999849i	\(-0.505531\pi\)
							0.857208	+	0.514971i	\(0.172197\pi\)
\(23\)	0.840363	−	3.13628i	0.175228	−	0.653959i	−0.821285	−	0.570518i	\(-0.806742\pi\)
							0.996513	−	0.0834407i	\(-0.0265909\pi\)
\(29\)	−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	\(0.136316\pi\)
\(37\)	−1.54277	+	5.75771i	−0.253631	+	0.946562i	0.715217	+	0.698903i	\(0.246328\pi\)
							−0.968847	+	0.247659i	\(0.920339\pi\)
\(41\)			5.84230i			0.912414i	0.889874	+	0.456207i	\(0.150792\pi\)
							−0.889874	+	0.456207i	\(0.849208\pi\)
\(43\)	−2.00369	+	2.00369i	−0.305559	+	0.305559i	−0.843184	−	0.537625i	\(-0.819322\pi\)
							0.537625	+	0.843184i	\(0.319322\pi\)
\(47\)	−5.10030	−	1.36662i	−0.743956	−	0.199342i	−0.133120	−	0.991100i	\(-0.542500\pi\)
							−0.610835	+	0.791758i	\(0.709166\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.2	✓	48	35.9	even	6
105.2.x.a.2.11	yes	48	105.44	odd	6
105.2.x.a.23.2	yes	48	35.2	odd	12
105.2.x.a.23.11	yes	48	105.2	even	12
105.2.x.a.32.2	yes	48	15.14	odd	2
105.2.x.a.32.11	yes	48	5.4	even	2
105.2.x.a.53.2	yes	48	15.2	even	4
105.2.x.a.53.11	yes	48	5.2	odd	4
525.2.bf.f.32.2		48	1.1	even	1	trivial
525.2.bf.f.32.11		48	3.2	odd	2	inner
525.2.bf.f.107.2		48	21.2	odd	6	inner
525.2.bf.f.107.11		48	7.2	even	3	inner
525.2.bf.f.368.2		48	5.3	odd	4	inner
525.2.bf.f.368.11		48	15.8	even	4	inner
525.2.bf.f.443.2		48	105.23	even	12	inner
525.2.bf.f.443.11		48	35.23	odd	12	inner
735.2.j.e.197.2		24	35.24	odd	6
735.2.j.e.197.11		24	105.59	even	6
735.2.j.e.638.2		24	105.17	odd	12
735.2.j.e.638.11		24	35.17	even	12
735.2.j.g.197.2		24	35.4	even	6
735.2.j.g.197.11		24	105.74	odd	6
735.2.j.g.638.2		24	105.32	even	12
735.2.j.g.638.11		24	35.32	odd	12
735.2.y.i.128.2		48	35.12	even	12
735.2.y.i.128.11		48	105.47	odd	12
735.2.y.i.263.2		48	105.62	odd	4
735.2.y.i.263.11		48	35.27	even	4
735.2.y.i.422.2		48	35.19	odd	6
735.2.y.i.422.11		48	105.89	even	6
735.2.y.i.557.2		48	105.104	even	2
735.2.y.i.557.11		48	35.34	odd	2