Embedded newform 105.2.p.a.59.2

• Label: 105.2.p.a.59.2
• 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			59.2
Character	\(\chi\)	\(=\)	105.59
Dual form			105.2.p.a.89.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25399 - 2.17197i) q^{2} +(1.51966 - 0.831052i) q^{3} +(-2.14497 + 3.71520i) q^{4} +(-1.93210 - 1.12560i) q^{5} +(-3.71065 - 2.25852i) q^{6} +(-1.65993 - 2.06025i) q^{7} +5.74313 q^{8} +(1.61871 - 2.52582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{4} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25399	−	2.17197i	−0.886704	−	1.53582i	−0.843749	−	0.536739i	\(-0.819656\pi\)
							−0.0429549	−	0.999077i	\(-0.513677\pi\)
\(3\)	1.51966	−	0.831052i	0.877374	−	0.479808i
							\(5\)	−1.93210	−	1.12560i	−0.864063	−	0.503383i
\(7\)	−1.65993	−	2.06025i	−0.627395	−	0.778701i
							\(11\)	1.48843	+	0.859346i	0.448779	+	0.259103i	0.707314	−	0.706899i	\(-0.249907\pi\)
−0.258535	+	0.966002i	\(0.583240\pi\)
\(13\)	0.360784			0.100063			0.0500317	−	0.998748i	\(-0.484068\pi\)
							0.0500317	+	0.998748i	\(0.484068\pi\)
\(17\)	2.20219	+	1.27144i	0.534110	+	0.308369i	0.742689	−	0.669637i	\(-0.233550\pi\)
							−0.208578	+	0.978006i	\(0.566884\pi\)
\(19\)	4.93492	−	2.84918i	1.13215	−	0.653646i	0.187674	−	0.982231i	\(-0.439905\pi\)
							0.944474	+	0.328586i	\(0.106572\pi\)
\(23\)	1.25399	+	2.17197i	0.261475	+	0.452887i	0.966634	−	0.256162i	\(-0.0824579\pi\)
							−0.705159	+	0.709049i	\(0.749125\pi\)
\(29\)		−	3.76228i		−	0.698638i	−0.937004	−	0.349319i	\(-0.886413\pi\)
							0.937004	−	0.349319i	\(-0.113587\pi\)
\(31\)	−2.41187	−	1.39249i	−0.433185	−	0.250099i	0.267518	−	0.963553i	\(-0.413797\pi\)
							−0.700702	+	0.713454i	\(0.747130\pi\)
\(37\)	−2.86279	+	1.65283i	−0.470639	+	0.271724i	−0.716507	−	0.697580i	\(-0.754260\pi\)
							0.245868	+	0.969303i	\(0.420927\pi\)
\(41\)	−2.63477			−0.411481			−0.205741	−	0.978607i	\(-0.565960\pi\)
							−0.205741	+	0.978607i	\(0.565960\pi\)
\(43\)			10.0606i			1.53423i	0.641510	+	0.767114i	\(0.278308\pi\)
							−0.641510	+	0.767114i	\(0.721692\pi\)
\(47\)	5.04290	−	2.91152i	0.735583	−	0.424689i	−0.0848783	−	0.996391i	\(-0.527050\pi\)
							0.820461	+	0.571702i	\(0.193717\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.p.a.59.2	yes	24
3.2	odd	2	inner	105.2.p.a.59.12	yes	24
5.2	odd	4		525.2.t.j.101.11		24
5.3	odd	4		525.2.t.j.101.2		24
5.4	even	2	inner	105.2.p.a.59.11	yes	24
7.2	even	3		735.2.p.f.509.2		24
7.3	odd	6		735.2.g.b.734.24		24
7.4	even	3		735.2.g.b.734.21		24
7.5	odd	6	inner	105.2.p.a.89.1	yes	24
7.6	odd	2		735.2.p.f.374.1		24
15.2	even	4		525.2.t.j.101.1		24
15.8	even	4		525.2.t.j.101.12		24
15.14	odd	2	inner	105.2.p.a.59.1	✓	24
21.2	odd	6		735.2.p.f.509.12		24
21.5	even	6	inner	105.2.p.a.89.11	yes	24
21.11	odd	6		735.2.g.b.734.2		24
21.17	even	6		735.2.g.b.734.3		24
21.20	even	2		735.2.p.f.374.11		24
35.4	even	6		735.2.g.b.734.4		24
35.9	even	6		735.2.p.f.509.11		24
35.12	even	12		525.2.t.j.26.1		24
35.19	odd	6	inner	105.2.p.a.89.12	yes	24
35.24	odd	6		735.2.g.b.734.1		24
35.33	even	12		525.2.t.j.26.12		24
35.34	odd	2		735.2.p.f.374.12		24
105.44	odd	6		735.2.p.f.509.1		24
105.47	odd	12		525.2.t.j.26.11		24
105.59	even	6		735.2.g.b.734.22		24
105.68	odd	12		525.2.t.j.26.2		24
105.74	odd	6		735.2.g.b.734.23		24
105.89	even	6	inner	105.2.p.a.89.2	yes	24
105.104	even	2		735.2.p.f.374.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.p.a.59.1	✓	24	15.14	odd	2	inner
105.2.p.a.59.2	yes	24	1.1	even	1	trivial
105.2.p.a.59.11	yes	24	5.4	even	2	inner
105.2.p.a.59.12	yes	24	3.2	odd	2	inner
105.2.p.a.89.1	yes	24	7.5	odd	6	inner
105.2.p.a.89.2	yes	24	105.89	even	6	inner
105.2.p.a.89.11	yes	24	21.5	even	6	inner
105.2.p.a.89.12	yes	24	35.19	odd	6	inner
525.2.t.j.26.1		24	35.12	even	12
525.2.t.j.26.2		24	105.68	odd	12
525.2.t.j.26.11		24	105.47	odd	12
525.2.t.j.26.12		24	35.33	even	12
525.2.t.j.101.1		24	15.2	even	4
525.2.t.j.101.2		24	5.3	odd	4
525.2.t.j.101.11		24	5.2	odd	4
525.2.t.j.101.12		24	15.8	even	4
735.2.g.b.734.1		24	35.24	odd	6
735.2.g.b.734.2		24	21.11	odd	6
735.2.g.b.734.3		24	21.17	even	6
735.2.g.b.734.4		24	35.4	even	6
735.2.g.b.734.21		24	7.4	even	3
735.2.g.b.734.22		24	105.59	even	6
735.2.g.b.734.23		24	105.74	odd	6
735.2.g.b.734.24		24	7.3	odd	6
735.2.p.f.374.1		24	7.6	odd	2
735.2.p.f.374.2		24	105.104	even	2
735.2.p.f.374.11		24	21.20	even	2
735.2.p.f.374.12		24	35.34	odd	2
735.2.p.f.509.1		24	105.44	odd	6
735.2.p.f.509.2		24	7.2	even	3
735.2.p.f.509.11		24	35.9	even	6
735.2.p.f.509.12		24	21.2	odd	6