Embedded newform 105.2.p.a.59.9

• Label: 105.2.p.a.59.9
• Level: $105$
• Weight: $2$
• Character: 105.59
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.9
Character	\(\chi\)	\(=\)	105.59
Dual form			105.2.p.a.89.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.757344 + 1.31176i) q^{2} +(-0.419611 + 1.68045i) q^{3} +(-0.147140 + 0.254854i) q^{4} +(-1.42830 + 1.72046i) q^{5} +(-2.52214 + 0.722254i) q^{6} +(-0.0753638 - 2.64468i) q^{7} +2.58363 q^{8} +(-2.64785 - 1.41027i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.757344	+	1.31176i	0.535523	+	0.927553i	0.999138	+	0.0415164i	\(0.0132189\pi\)
							−0.463615	+	0.886037i	\(0.653448\pi\)
\(3\)	−0.419611	+	1.68045i	−0.242263	+	0.970211i
							\(5\)	−1.42830	+	1.72046i	−0.638754	+	0.769411i
\(7\)	−0.0753638	−	2.64468i	−0.0284849	−	0.999594i
							\(11\)	1.86048	+	1.07415i	0.560957	+	0.323869i	0.753529	−	0.657414i	\(-0.228350\pi\)
−0.192573	+	0.981283i	\(0.561683\pi\)
\(13\)	3.48097			0.965448			0.482724	−	0.875773i	\(-0.339647\pi\)
							0.482724	+	0.875773i	\(0.339647\pi\)
\(17\)	−3.09793	−	1.78859i	−0.751359	−	0.433797i	0.0748259	−	0.997197i	\(-0.476160\pi\)
							−0.826185	+	0.563399i	\(0.809493\pi\)
\(19\)	−1.05858	+	0.611171i	−0.242855	+	0.140212i	−0.616488	−	0.787364i	\(-0.711445\pi\)
							0.373633	+	0.927576i	\(0.378112\pi\)
\(23\)	−0.757344	−	1.31176i	−0.157917	−	0.273521i	0.776200	−	0.630486i	\(-0.217145\pi\)
							−0.934117	+	0.356966i	\(0.883811\pi\)
\(29\)		−	5.95645i		−	1.10608i	−0.833153	−	0.553042i	\(-0.813467\pi\)
							0.833153	−	0.553042i	\(-0.186533\pi\)
\(31\)	2.75098	+	1.58828i	0.494091	+	0.285263i	0.726270	−	0.687410i	\(-0.241252\pi\)
							−0.232179	+	0.972673i	\(0.574586\pi\)
\(37\)	−6.75803	+	3.90175i	−1.11101	+	0.641444i	−0.939092	−	0.343667i	\(-0.888331\pi\)
							−0.171921	+	0.985111i	\(0.554998\pi\)
\(41\)	−11.8685			−1.85355			−0.926773	−	0.375622i	\(-0.877429\pi\)
							−0.926773	+	0.375622i	\(0.877429\pi\)
\(43\)		−	2.99294i		−	0.456419i	−0.973612	−	0.228209i	\(-0.926713\pi\)
							0.973612	−	0.228209i	\(-0.0732871\pi\)
\(47\)	5.28420	−	3.05084i	0.770780	−	0.445010i	−0.0623727	−	0.998053i	\(-0.519867\pi\)
							0.833153	+	0.553043i	\(0.186533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.p.a.59.9	yes	24
3.2	odd	2	inner	105.2.p.a.59.3	✓	24
5.2	odd	4		525.2.t.j.101.4		24
5.3	odd	4		525.2.t.j.101.9		24
5.4	even	2	inner	105.2.p.a.59.4	yes	24
7.2	even	3		735.2.p.f.509.9		24
7.3	odd	6		735.2.g.b.734.6		24
7.4	even	3		735.2.g.b.734.7		24
7.5	odd	6	inner	105.2.p.a.89.10	yes	24
7.6	odd	2		735.2.p.f.374.10		24
15.2	even	4		525.2.t.j.101.10		24
15.8	even	4		525.2.t.j.101.3		24
15.14	odd	2	inner	105.2.p.a.59.10	yes	24
21.2	odd	6		735.2.p.f.509.3		24
21.5	even	6	inner	105.2.p.a.89.4	yes	24
21.11	odd	6		735.2.g.b.734.20		24
21.17	even	6		735.2.g.b.734.17		24
21.20	even	2		735.2.p.f.374.4		24
35.4	even	6		735.2.g.b.734.18		24
35.9	even	6		735.2.p.f.509.4		24
35.12	even	12		525.2.t.j.26.10		24
35.19	odd	6	inner	105.2.p.a.89.3	yes	24
35.24	odd	6		735.2.g.b.734.19		24
35.33	even	12		525.2.t.j.26.3		24
35.34	odd	2		735.2.p.f.374.3		24
105.44	odd	6		735.2.p.f.509.10		24
105.47	odd	12		525.2.t.j.26.4		24
105.59	even	6		735.2.g.b.734.8		24
105.68	odd	12		525.2.t.j.26.9		24
105.74	odd	6		735.2.g.b.734.5		24
105.89	even	6	inner	105.2.p.a.89.9	yes	24
105.104	even	2		735.2.p.f.374.9		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.p.a.59.3	✓	24	3.2	odd	2	inner
105.2.p.a.59.4	yes	24	5.4	even	2	inner
105.2.p.a.59.9	yes	24	1.1	even	1	trivial
105.2.p.a.59.10	yes	24	15.14	odd	2	inner
105.2.p.a.89.3	yes	24	35.19	odd	6	inner
105.2.p.a.89.4	yes	24	21.5	even	6	inner
105.2.p.a.89.9	yes	24	105.89	even	6	inner
105.2.p.a.89.10	yes	24	7.5	odd	6	inner
525.2.t.j.26.3		24	35.33	even	12
525.2.t.j.26.4		24	105.47	odd	12
525.2.t.j.26.9		24	105.68	odd	12
525.2.t.j.26.10		24	35.12	even	12
525.2.t.j.101.3		24	15.8	even	4
525.2.t.j.101.4		24	5.2	odd	4
525.2.t.j.101.9		24	5.3	odd	4
525.2.t.j.101.10		24	15.2	even	4
735.2.g.b.734.5		24	105.74	odd	6
735.2.g.b.734.6		24	7.3	odd	6
735.2.g.b.734.7		24	7.4	even	3
735.2.g.b.734.8		24	105.59	even	6
735.2.g.b.734.17		24	21.17	even	6
735.2.g.b.734.18		24	35.4	even	6
735.2.g.b.734.19		24	35.24	odd	6
735.2.g.b.734.20		24	21.11	odd	6
735.2.p.f.374.3		24	35.34	odd	2
735.2.p.f.374.4		24	21.20	even	2
735.2.p.f.374.9		24	105.104	even	2
735.2.p.f.374.10		24	7.6	odd	2
735.2.p.f.509.3		24	21.2	odd	6
735.2.p.f.509.4		24	35.9	even	6
735.2.p.f.509.9		24	7.2	even	3
735.2.p.f.509.10		24	105.44	odd	6