Embedded newform 105.2.p.a.59.5

• Label: 105.2.p.a.59.5
• 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			59.5
Character	\(\chi\)	\(=\)	105.59
Dual form			105.2.p.a.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.322403 - 0.558418i) q^{2} +(1.15761 + 1.28838i) q^{3} +(0.792113 - 1.37198i) q^{4} +(-1.60193 - 1.56008i) q^{5} +(0.346239 - 1.06181i) q^{6} +(2.64366 + 0.105130i) q^{7} -2.31113 q^{8} +(-0.319861 + 2.98290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{4} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.322403	−	0.558418i	−0.227973	−	0.394861i	0.729234	−	0.684264i	\(-0.239877\pi\)
							−0.957207	+	0.289403i	\(0.906543\pi\)
\(3\)	1.15761	+	1.28838i	0.668349	+	0.743848i
							\(5\)	−1.60193	−	1.56008i	−0.716403	−	0.697687i
\(7\)	2.64366	+	0.105130i	0.999210	+	0.0397354i
							\(11\)	3.51044	+	2.02675i	1.05844	+	0.611089i	0.924999	−	0.379968i	\(-0.124065\pi\)
0.133437	+	0.991057i	\(0.457398\pi\)
\(13\)	−4.21339			−1.16858			−0.584292	−	0.811543i	\(-0.698628\pi\)
							−0.584292	+	0.811543i	\(0.698628\pi\)
\(17\)	−1.88498	−	1.08830i	−0.457176	−	0.263951i	0.253680	−	0.967288i	\(-0.418359\pi\)
							−0.710856	+	0.703338i	\(0.751692\pi\)
\(19\)	−3.87634	+	2.23800i	−0.889293	+	0.513434i	−0.873711	−	0.486445i	\(-0.838293\pi\)
							−0.0155818	+	0.999879i	\(0.504960\pi\)
\(23\)	0.322403	+	0.558418i	0.0672257	+	0.116438i	0.897679	−	0.440650i	\(-0.145252\pi\)
							−0.830453	+	0.557088i	\(0.811919\pi\)
\(29\)		−	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	\(0.677862\pi\)
\(37\)	−3.69236	+	2.13178i	−0.607020	+	0.350463i	−0.771798	−	0.635868i	\(-0.780642\pi\)
							0.164778	+	0.986331i	\(0.447309\pi\)
\(41\)	2.27971			0.356031			0.178016	−	0.984028i	\(-0.443032\pi\)
							0.178016	+	0.984028i	\(0.443032\pi\)
\(43\)		−	6.54419i		−	0.997980i	−0.866608	−	0.498990i	\(-0.833705\pi\)
							0.866608	−	0.498990i	\(-0.166295\pi\)
\(47\)	6.75621	−	3.90070i	0.985494	−	0.568975i	0.0815698	−	0.996668i	\(-0.474007\pi\)
							0.903924	+	0.427692i	\(0.140673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.p.a.59.5	✓	24
3.2	odd	2	inner	105.2.p.a.59.7	yes	24
5.2	odd	4		525.2.t.j.101.8		24
5.3	odd	4		525.2.t.j.101.5		24
5.4	even	2	inner	105.2.p.a.59.8	yes	24
7.2	even	3		735.2.p.f.509.5		24
7.3	odd	6		735.2.g.b.734.14		24
7.4	even	3		735.2.g.b.734.15		24
7.5	odd	6	inner	105.2.p.a.89.6	yes	24
7.6	odd	2		735.2.p.f.374.6		24
15.2	even	4		525.2.t.j.101.6		24
15.8	even	4		525.2.t.j.101.7		24
15.14	odd	2	inner	105.2.p.a.59.6	yes	24
21.2	odd	6		735.2.p.f.509.7		24
21.5	even	6	inner	105.2.p.a.89.8	yes	24
21.11	odd	6		735.2.g.b.734.12		24
21.17	even	6		735.2.g.b.734.9		24
21.20	even	2		735.2.p.f.374.8		24
35.4	even	6		735.2.g.b.734.10		24
35.9	even	6		735.2.p.f.509.8		24
35.12	even	12		525.2.t.j.26.6		24
35.19	odd	6	inner	105.2.p.a.89.7	yes	24
35.24	odd	6		735.2.g.b.734.11		24
35.33	even	12		525.2.t.j.26.7		24
35.34	odd	2		735.2.p.f.374.7		24
105.44	odd	6		735.2.p.f.509.6		24
105.47	odd	12		525.2.t.j.26.8		24
105.59	even	6		735.2.g.b.734.16		24
105.68	odd	12		525.2.t.j.26.5		24
105.74	odd	6		735.2.g.b.734.13		24
105.89	even	6	inner	105.2.p.a.89.5	yes	24
105.104	even	2		735.2.p.f.374.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.p.a.59.5	✓	24	1.1	even	1	trivial
105.2.p.a.59.6	yes	24	15.14	odd	2	inner
105.2.p.a.59.7	yes	24	3.2	odd	2	inner
105.2.p.a.59.8	yes	24	5.4	even	2	inner
105.2.p.a.89.5	yes	24	105.89	even	6	inner
105.2.p.a.89.6	yes	24	7.5	odd	6	inner
105.2.p.a.89.7	yes	24	35.19	odd	6	inner
105.2.p.a.89.8	yes	24	21.5	even	6	inner
525.2.t.j.26.5		24	105.68	odd	12
525.2.t.j.26.6		24	35.12	even	12
525.2.t.j.26.7		24	35.33	even	12
525.2.t.j.26.8		24	105.47	odd	12
525.2.t.j.101.5		24	5.3	odd	4
525.2.t.j.101.6		24	15.2	even	4
525.2.t.j.101.7		24	15.8	even	4
525.2.t.j.101.8		24	5.2	odd	4
735.2.g.b.734.9		24	21.17	even	6
735.2.g.b.734.10		24	35.4	even	6
735.2.g.b.734.11		24	35.24	odd	6
735.2.g.b.734.12		24	21.11	odd	6
735.2.g.b.734.13		24	105.74	odd	6
735.2.g.b.734.14		24	7.3	odd	6
735.2.g.b.734.15		24	7.4	even	3
735.2.g.b.734.16		24	105.59	even	6
735.2.p.f.374.5		24	105.104	even	2
735.2.p.f.374.6		24	7.6	odd	2
735.2.p.f.374.7		24	35.34	odd	2
735.2.p.f.374.8		24	21.20	even	2
735.2.p.f.509.5		24	7.2	even	3
735.2.p.f.509.6		24	105.44	odd	6
735.2.p.f.509.7		24	21.2	odd	6
735.2.p.f.509.8		24	35.9	even	6