Embedded newform 525.2.t.j.101.7

• Label: 525.2.t.j.101.7
• Level: $525$
• Weight: $2$
• Character: 525.101
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.7
Character	\(\chi\)	\(=\)	525.101
Dual form			525.2.t.j.26.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.558418 - 0.322403i) q^{2} +(0.358331 - 1.69458i) q^{3} +(-0.792113 + 1.37198i) q^{4} +(-0.346239 - 1.06181i) q^{6} +(0.105130 - 2.64366i) q^{7} +2.31113i q^{8} +(-2.74320 - 1.21444i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 12 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.558418	−	0.322403i	0.394861	−	0.227973i	−0.289403	−	0.957207i	\(-0.593457\pi\)
							0.684264	+	0.729234i	\(0.260123\pi\)
\(3\)	0.358331	−	1.69458i	0.206883	−	0.978366i
							\(5\)		0			0
\(7\)	0.105130	−	2.64366i	0.0397354	−	0.999210i
							\(11\)	−3.51044	−	2.02675i	−1.05844	−	0.611089i	−0.133437	−	0.991057i	\(-0.542602\pi\)
−0.924999	+	0.379968i	\(0.875935\pi\)
\(13\)		−	4.21339i		−	1.16858i	−0.811543	−	0.584292i	\(-0.801372\pi\)
							0.811543	−	0.584292i	\(-0.198628\pi\)
\(17\)	1.08830	−	1.88498i	0.263951	−	0.457176i	−0.703338	−	0.710856i	\(-0.748308\pi\)
							0.967288	+	0.253680i	\(0.0816411\pi\)
\(19\)	3.87634	−	2.23800i	0.889293	−	0.513434i	0.0155818	−	0.999879i	\(-0.495040\pi\)
							0.873711	+	0.486445i	\(0.161707\pi\)
\(23\)	0.558418	−	0.322403i	0.116438	−	0.0672257i	−0.440650	−	0.897679i	\(-0.645252\pi\)
							0.557088	+	0.830453i	\(0.311919\pi\)
\(29\)		−	1.16875i		−	0.217032i	−0.994095	−	0.108516i	\(-0.965390\pi\)
							0.994095	−	0.108516i	\(-0.0346099\pi\)
\(31\)	−0.339111	−	0.195786i	−0.0609061	−	0.0351641i	0.469238	−	0.883072i	\(-0.344529\pi\)
							−0.530144	+	0.847908i	\(0.677862\pi\)
\(37\)	2.13178	+	3.69236i	0.350463	+	0.607020i	0.986331	−	0.164778i	\(-0.0526909\pi\)
							−0.635868	+	0.771798i	\(0.719358\pi\)
\(41\)	−2.27971			−0.356031			−0.178016	−	0.984028i	\(-0.556968\pi\)
							−0.178016	+	0.984028i	\(0.556968\pi\)
\(43\)	6.54419			0.997980			0.498990	−	0.866608i	\(-0.333705\pi\)
							0.498990	+	0.866608i	\(0.333705\pi\)
\(47\)	3.90070	+	6.75621i	0.568975	+	0.985494i	0.996668	+	0.0815698i	\(0.0259934\pi\)
							−0.427692	+	0.903924i	\(0.640673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.t.j.101.7		24
3.2	odd	2	inner	525.2.t.j.101.5		24
5.2	odd	4		105.2.p.a.59.7	yes	24
5.3	odd	4		105.2.p.a.59.6	yes	24
5.4	even	2	inner	525.2.t.j.101.6		24
7.5	odd	6	inner	525.2.t.j.26.5		24
15.2	even	4		105.2.p.a.59.5	✓	24
15.8	even	4		105.2.p.a.59.8	yes	24
15.14	odd	2	inner	525.2.t.j.101.8		24
21.5	even	6	inner	525.2.t.j.26.7		24
35.2	odd	12		735.2.p.f.509.7		24
35.3	even	12		735.2.g.b.734.16		24
35.12	even	12		105.2.p.a.89.8	yes	24
35.13	even	4		735.2.p.f.374.5		24
35.17	even	12		735.2.g.b.734.9		24
35.18	odd	12		735.2.g.b.734.13		24
35.19	odd	6	inner	525.2.t.j.26.8		24
35.23	odd	12		735.2.p.f.509.6		24
35.27	even	4		735.2.p.f.374.8		24
35.32	odd	12		735.2.g.b.734.12		24
35.33	even	12		105.2.p.a.89.5	yes	24
105.2	even	12		735.2.p.f.509.5		24
105.17	odd	12		735.2.g.b.734.14		24
105.23	even	12		735.2.p.f.509.8		24
105.32	even	12		735.2.g.b.734.15		24
105.38	odd	12		735.2.g.b.734.11		24
105.47	odd	12		105.2.p.a.89.6	yes	24
105.53	even	12		735.2.g.b.734.10		24
105.62	odd	4		735.2.p.f.374.6		24
105.68	odd	12		105.2.p.a.89.7	yes	24
105.83	odd	4		735.2.p.f.374.7		24
105.89	even	6	inner	525.2.t.j.26.6		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.p.a.59.5	✓	24	15.2	even	4
105.2.p.a.59.6	yes	24	5.3	odd	4
105.2.p.a.59.7	yes	24	5.2	odd	4
105.2.p.a.59.8	yes	24	15.8	even	4
105.2.p.a.89.5	yes	24	35.33	even	12
105.2.p.a.89.6	yes	24	105.47	odd	12
105.2.p.a.89.7	yes	24	105.68	odd	12
105.2.p.a.89.8	yes	24	35.12	even	12
525.2.t.j.26.5		24	7.5	odd	6	inner
525.2.t.j.26.6		24	105.89	even	6	inner
525.2.t.j.26.7		24	21.5	even	6	inner
525.2.t.j.26.8		24	35.19	odd	6	inner
525.2.t.j.101.5		24	3.2	odd	2	inner
525.2.t.j.101.6		24	5.4	even	2	inner
525.2.t.j.101.7		24	1.1	even	1	trivial
525.2.t.j.101.8		24	15.14	odd	2	inner
735.2.g.b.734.9		24	35.17	even	12
735.2.g.b.734.10		24	105.53	even	12
735.2.g.b.734.11		24	105.38	odd	12
735.2.g.b.734.12		24	35.32	odd	12
735.2.g.b.734.13		24	35.18	odd	12
735.2.g.b.734.14		24	105.17	odd	12
735.2.g.b.734.15		24	105.32	even	12
735.2.g.b.734.16		24	35.3	even	12
735.2.p.f.374.5		24	35.13	even	4
735.2.p.f.374.6		24	105.62	odd	4
735.2.p.f.374.7		24	105.83	odd	4
735.2.p.f.374.8		24	35.27	even	4
735.2.p.f.509.5		24	105.2	even	12
735.2.p.f.509.6		24	35.23	odd	12
735.2.p.f.509.7		24	35.2	odd	12
735.2.p.f.509.8		24	105.23	even	12