Embedded newform 525.2.bf.f.32.4

• Label: 525.2.bf.f.32.4
• 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			32.4
Character	\(\chi\)	\(=\)	525.32
Dual form			525.2.bf.f.443.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26950 - 0.340162i) q^{2} +(-1.71749 + 0.224146i) q^{3} +(-0.236127 - 0.136328i) q^{4} +(2.25660 + 0.299670i) q^{6} +(-1.25943 - 2.32676i) q^{7} +(2.11207 + 2.11207i) q^{8} +(2.89952 - 0.769934i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} - 24 q^{6} + 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26950	−	0.340162i	−0.897673	−	0.240531i	−0.219656	−	0.975577i	\(-0.570494\pi\)
							−0.678017	+	0.735046i	\(0.737160\pi\)
\(3\)	−1.71749	+	0.224146i	−0.991591	+	0.129411i
							\(5\)		0			0
\(7\)	−1.25943	−	2.32676i	−0.476021	−	0.879434i
							\(11\)	−3.38224	−	1.95274i	−1.01978	−	0.588773i	−0.105743	−	0.994394i	\(-0.533722\pi\)
−0.914041	+	0.405621i	\(0.867055\pi\)
\(13\)	1.56642	−	1.56642i	0.434448	−	0.434448i	−0.455691	−	0.890138i	\(-0.650608\pi\)
							0.890138	+	0.455691i	\(0.150608\pi\)
\(17\)	−0.693065	−	2.58656i	−0.168093	−	0.627332i	−0.997625	−	0.0688731i	\(-0.978060\pi\)
							0.829532	−	0.558459i	\(-0.188607\pi\)
\(19\)	1.61097	−	0.930096i	0.369582	−	0.213379i	−0.303694	−	0.952770i	\(-0.598220\pi\)
							0.673276	+	0.739391i	\(0.264887\pi\)
\(23\)	−0.638564	+	2.38315i	−0.133150	+	0.496921i	−0.999999	−	0.00164943i	\(-0.999475\pi\)
							0.866849	+	0.498571i	\(0.166142\pi\)
\(29\)	0.513153			0.0952901			0.0476450	−	0.998864i	\(-0.484828\pi\)
							0.0476450	+	0.998864i	\(0.484828\pi\)
\(31\)	−4.29138	+	7.43289i	−0.770755	+	1.33499i	0.166394	+	0.986059i	\(0.446788\pi\)
							−0.937150	+	0.348928i	\(0.886546\pi\)
\(37\)	−1.77034	+	6.60698i	−0.291041	+	1.08618i	0.653269	+	0.757126i	\(0.273397\pi\)
							−0.944311	+	0.329056i	\(0.893270\pi\)
\(41\)			0.308469i			0.0481748i	0.999710	+	0.0240874i	\(0.00766800\pi\)
							−0.999710	+	0.0240874i	\(0.992332\pi\)
\(43\)	−7.60892	+	7.60892i	−1.16035	+	1.16035i	−0.175950	+	0.984399i	\(0.556300\pi\)
							−0.984399	+	0.175950i	\(0.943700\pi\)
\(47\)	−5.10994	−	1.36920i	−0.745361	−	0.199719i	−0.133902	−	0.990995i	\(-0.542751\pi\)
							−0.611460	+	0.791276i	\(0.709417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bf.f.32.4		48
3.2	odd	2	inner	525.2.bf.f.32.9		48
5.2	odd	4		105.2.x.a.53.9	yes	48
5.3	odd	4	inner	525.2.bf.f.368.4		48
5.4	even	2		105.2.x.a.32.9	yes	48
7.2	even	3	inner	525.2.bf.f.107.9		48
15.2	even	4		105.2.x.a.53.4	yes	48
15.8	even	4	inner	525.2.bf.f.368.9		48
15.14	odd	2		105.2.x.a.32.4	yes	48
21.2	odd	6	inner	525.2.bf.f.107.4		48
35.2	odd	12		105.2.x.a.23.4	yes	48
35.4	even	6		735.2.j.g.197.4		24
35.9	even	6		105.2.x.a.2.4	✓	48
35.12	even	12		735.2.y.i.128.4		48
35.17	even	12		735.2.j.e.638.9		24
35.19	odd	6		735.2.y.i.422.4		48
35.23	odd	12	inner	525.2.bf.f.443.9		48
35.24	odd	6		735.2.j.e.197.4		24
35.27	even	4		735.2.y.i.263.9		48
35.32	odd	12		735.2.j.g.638.9		24
35.34	odd	2		735.2.y.i.557.9		48
105.2	even	12		105.2.x.a.23.9	yes	48
105.17	odd	12		735.2.j.e.638.4		24
105.23	even	12	inner	525.2.bf.f.443.4		48
105.32	even	12		735.2.j.g.638.4		24
105.44	odd	6		105.2.x.a.2.9	yes	48
105.47	odd	12		735.2.y.i.128.9		48
105.59	even	6		735.2.j.e.197.9		24
105.62	odd	4		735.2.y.i.263.4		48
105.74	odd	6		735.2.j.g.197.9		24
105.89	even	6		735.2.y.i.422.9		48
105.104	even	2		735.2.y.i.557.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.4	✓	48	35.9	even	6
105.2.x.a.2.9	yes	48	105.44	odd	6
105.2.x.a.23.4	yes	48	35.2	odd	12
105.2.x.a.23.9	yes	48	105.2	even	12
105.2.x.a.32.4	yes	48	15.14	odd	2
105.2.x.a.32.9	yes	48	5.4	even	2
105.2.x.a.53.4	yes	48	15.2	even	4
105.2.x.a.53.9	yes	48	5.2	odd	4
525.2.bf.f.32.4		48	1.1	even	1	trivial
525.2.bf.f.32.9		48	3.2	odd	2	inner
525.2.bf.f.107.4		48	21.2	odd	6	inner
525.2.bf.f.107.9		48	7.2	even	3	inner
525.2.bf.f.368.4		48	5.3	odd	4	inner
525.2.bf.f.368.9		48	15.8	even	4	inner
525.2.bf.f.443.4		48	105.23	even	12	inner
525.2.bf.f.443.9		48	35.23	odd	12	inner
735.2.j.e.197.4		24	35.24	odd	6
735.2.j.e.197.9		24	105.59	even	6
735.2.j.e.638.4		24	105.17	odd	12
735.2.j.e.638.9		24	35.17	even	12
735.2.j.g.197.4		24	35.4	even	6
735.2.j.g.197.9		24	105.74	odd	6
735.2.j.g.638.4		24	105.32	even	12
735.2.j.g.638.9		24	35.32	odd	12
735.2.y.i.128.4		48	35.12	even	12
735.2.y.i.128.9		48	105.47	odd	12
735.2.y.i.263.4		48	105.62	odd	4
735.2.y.i.263.9		48	35.27	even	4
735.2.y.i.422.4		48	35.19	odd	6
735.2.y.i.422.9		48	105.89	even	6
735.2.y.i.557.4		48	105.104	even	2
735.2.y.i.557.9		48	35.34	odd	2