Embedded newform 525.2.bf.f.107.1

• Label: 525.2.bf.f.107.1
• Level: $525$
• Weight: $2$
• Character: 525.107
• 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			107.1
Character	\(\chi\)	\(=\)	525.107
Dual form			525.2.bf.f.368.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.631395 - 2.35640i) q^{2} +(0.775426 - 1.54878i) q^{3} +(-3.42191 + 1.97564i) q^{4} +(-4.13914 - 0.849321i) q^{6} +(-1.82148 + 1.91891i) q^{7} +(3.36596 + 3.36596i) q^{8} +(-1.79743 - 2.40193i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} - 24 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.631395	−	2.35640i	−0.446464	−	1.66623i	−0.712042	−	0.702137i	\(-0.752230\pi\)
							0.265578	−	0.964089i	\(-0.414437\pi\)
\(3\)	0.775426	−	1.54878i	0.447692	−	0.894188i
							\(5\)		0			0
\(7\)	−1.82148	+	1.91891i	−0.688453	+	0.725281i
							\(11\)	−3.08053	+	1.77855i	−0.928816	+	0.536252i	−0.886437	−	0.462850i	\(-0.846827\pi\)
−0.0423788	+	0.999102i	\(0.513494\pi\)
\(13\)	−1.28412	+	1.28412i	−0.356151	+	0.356151i	−0.862392	−	0.506241i	\(-0.831035\pi\)
							0.506241	+	0.862392i	\(0.331035\pi\)
\(17\)	−2.95633	−	0.792145i	−0.717015	−	0.192123i	−0.118175	−	0.992993i	\(-0.537704\pi\)
							−0.598839	+	0.800869i	\(0.704371\pi\)
\(19\)	−0.331717	−	0.191517i	−0.0761011	−	0.0439370i	0.461467	−	0.887158i	\(-0.347323\pi\)
							−0.537568	+	0.843221i	\(0.680657\pi\)
\(23\)	2.45814	−	0.658656i	0.512557	−	0.137339i	0.00673550	−	0.999977i	\(-0.497856\pi\)
							0.505822	+	0.862638i	\(0.331189\pi\)
\(29\)	−5.51741			−1.02456			−0.512279	−	0.858819i	\(-0.671199\pi\)
							−0.512279	+	0.858819i	\(0.671199\pi\)
\(31\)	0.323980	+	0.561149i	0.0581885	+	0.100785i	0.893652	−	0.448760i	\(-0.148134\pi\)
							−0.835464	+	0.549546i	\(0.814801\pi\)
\(37\)	−5.00473	+	1.34101i	−0.822772	+	0.220461i	−0.645558	−	0.763711i	\(-0.723375\pi\)
							−0.177214	+	0.984172i	\(0.556709\pi\)
\(41\)		−	10.1075i		−	1.57852i	−0.614060	−	0.789259i	\(-0.710465\pi\)
							0.614060	−	0.789259i	\(-0.289535\pi\)
\(43\)	0.335236	−	0.335236i	0.0511231	−	0.0511231i	−0.681083	−	0.732206i	\(-0.738491\pi\)
							0.732206	+	0.681083i	\(0.238491\pi\)
\(47\)	−0.751687	−	2.80533i	−0.109645	−	0.409200i	0.889186	−	0.457546i	\(-0.151272\pi\)
							−0.998831	+	0.0483463i	\(0.984605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bf.f.107.1		48
3.2	odd	2	inner	525.2.bf.f.107.12		48
5.2	odd	4		105.2.x.a.23.12	yes	48
5.3	odd	4	inner	525.2.bf.f.443.1		48
5.4	even	2		105.2.x.a.2.12	yes	48
7.4	even	3	inner	525.2.bf.f.32.12		48
15.2	even	4		105.2.x.a.23.1	yes	48
15.8	even	4	inner	525.2.bf.f.443.12		48
15.14	odd	2		105.2.x.a.2.1	✓	48
21.11	odd	6	inner	525.2.bf.f.32.1		48
35.2	odd	12		735.2.j.g.638.1		24
35.4	even	6		105.2.x.a.32.1	yes	48
35.9	even	6		735.2.j.g.197.12		24
35.12	even	12		735.2.j.e.638.1		24
35.17	even	12		735.2.y.i.263.1		48
35.18	odd	12	inner	525.2.bf.f.368.12		48
35.19	odd	6		735.2.j.e.197.12		24
35.24	odd	6		735.2.y.i.557.1		48
35.27	even	4		735.2.y.i.128.12		48
35.32	odd	12		105.2.x.a.53.1	yes	48
35.34	odd	2		735.2.y.i.422.12		48
105.2	even	12		735.2.j.g.638.12		24
105.17	odd	12		735.2.y.i.263.12		48
105.32	even	12		105.2.x.a.53.12	yes	48
105.44	odd	6		735.2.j.g.197.1		24
105.47	odd	12		735.2.j.e.638.12		24
105.53	even	12	inner	525.2.bf.f.368.1		48
105.59	even	6		735.2.y.i.557.12		48
105.62	odd	4		735.2.y.i.128.1		48
105.74	odd	6		105.2.x.a.32.12	yes	48
105.89	even	6		735.2.j.e.197.1		24
105.104	even	2		735.2.y.i.422.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.1	✓	48	15.14	odd	2
105.2.x.a.2.12	yes	48	5.4	even	2
105.2.x.a.23.1	yes	48	15.2	even	4
105.2.x.a.23.12	yes	48	5.2	odd	4
105.2.x.a.32.1	yes	48	35.4	even	6
105.2.x.a.32.12	yes	48	105.74	odd	6
105.2.x.a.53.1	yes	48	35.32	odd	12
105.2.x.a.53.12	yes	48	105.32	even	12
525.2.bf.f.32.1		48	21.11	odd	6	inner
525.2.bf.f.32.12		48	7.4	even	3	inner
525.2.bf.f.107.1		48	1.1	even	1	trivial
525.2.bf.f.107.12		48	3.2	odd	2	inner
525.2.bf.f.368.1		48	105.53	even	12	inner
525.2.bf.f.368.12		48	35.18	odd	12	inner
525.2.bf.f.443.1		48	5.3	odd	4	inner
525.2.bf.f.443.12		48	15.8	even	4	inner
735.2.j.e.197.1		24	105.89	even	6
735.2.j.e.197.12		24	35.19	odd	6
735.2.j.e.638.1		24	35.12	even	12
735.2.j.e.638.12		24	105.47	odd	12
735.2.j.g.197.1		24	105.44	odd	6
735.2.j.g.197.12		24	35.9	even	6
735.2.j.g.638.1		24	35.2	odd	12
735.2.j.g.638.12		24	105.2	even	12
735.2.y.i.128.1		48	105.62	odd	4
735.2.y.i.128.12		48	35.27	even	4
735.2.y.i.263.1		48	35.17	even	12
735.2.y.i.263.12		48	105.17	odd	12
735.2.y.i.422.1		48	105.104	even	2
735.2.y.i.422.12		48	35.34	odd	2
735.2.y.i.557.1		48	35.24	odd	6
735.2.y.i.557.12		48	105.59	even	6