Embedded newform 105.2.x.a.53.6

• Label: 105.2.x.a.53.6
• Level: $105$
• Weight: $2$
• Character: 105.53
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			53.6
Character	\(\chi\)	\(=\)	105.53
Dual form			105.2.x.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0799329 + 0.298314i) q^{2} +(1.29105 + 1.15464i) q^{3} +(1.64945 + 0.952310i) q^{4} +(-0.596180 - 2.15513i) q^{5} +(-0.447643 + 0.292843i) q^{6} +(-2.46856 - 0.951942i) q^{7} +(-0.852694 + 0.852694i) q^{8} +(0.333606 + 2.98139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0799329	+	0.298314i	−0.0565211	+	0.210940i	−0.988411	−	0.151802i	\(-0.951492\pi\)
							0.931890	+	0.362741i	\(0.118159\pi\)
\(3\)	1.29105	+	1.15464i	0.745386	+	0.666633i
							\(5\)	−0.596180	−	2.15513i	−0.266620	−	0.963802i
\(7\)	−2.46856	−	0.951942i	−0.933029	−	0.359800i
							\(11\)	−0.660315	−	0.381233i	−0.199092	−	0.114946i	0.397140	−	0.917758i	\(-0.370003\pi\)
−0.596232	+	0.802812i	\(0.703336\pi\)
\(13\)	−2.27077	−	2.27077i	−0.629797	−	0.629797i	0.318220	−	0.948017i	\(-0.396915\pi\)
							−0.948017	+	0.318220i	\(0.896915\pi\)
\(17\)	4.69471	−	1.25794i	1.13864	−	0.305096i	0.360233	−	0.932862i	\(-0.382697\pi\)
							0.778402	+	0.627766i	\(0.216030\pi\)
\(19\)	−1.41761	+	0.818455i	−0.325221	+	0.187767i	−0.653717	−	0.756739i	\(-0.726791\pi\)
							0.328496	+	0.944505i	\(0.393458\pi\)
\(23\)	−7.39003	−	1.98015i	−1.54093	−	0.412890i	−0.614363	−	0.789024i	\(-0.710587\pi\)
							−0.926566	+	0.376133i	\(0.877253\pi\)
\(29\)	4.94251			0.917801			0.458900	−	0.888488i	\(-0.348243\pi\)
							0.458900	+	0.888488i	\(0.348243\pi\)
\(31\)	2.96413	−	5.13403i	0.532374	−	0.922099i	−0.466911	−	0.884304i	\(-0.654633\pi\)
							0.999286	−	0.0377949i	\(-0.0120334\pi\)
\(37\)	3.41587	+	0.915280i	0.561566	+	0.150471i	0.528426	−	0.848980i	\(-0.322783\pi\)
							0.0331401	+	0.999451i	\(0.489449\pi\)
\(41\)		−	4.35963i		−	0.680860i	−0.940270	−	0.340430i	\(-0.889427\pi\)
							0.940270	−	0.340430i	\(-0.110573\pi\)
\(43\)	2.69037	+	2.69037i	0.410277	+	0.410277i	0.881835	−	0.471558i	\(-0.156308\pi\)
							−0.471558	+	0.881835i	\(0.656308\pi\)
\(47\)	−1.10971	+	4.14148i	−0.161867	+	0.604097i	0.836552	+	0.547888i	\(0.184568\pi\)
							−0.998419	+	0.0562089i	\(0.982099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.x.a.53.6	yes	48
3.2	odd	2	inner	105.2.x.a.53.7	yes	48
5.2	odd	4	inner	105.2.x.a.32.6	yes	48
5.3	odd	4		525.2.bf.f.32.7		48
5.4	even	2		525.2.bf.f.368.7		48
7.2	even	3	inner	105.2.x.a.23.7	yes	48
7.3	odd	6		735.2.j.e.638.6		24
7.4	even	3		735.2.j.g.638.6		24
7.5	odd	6		735.2.y.i.128.7		48
7.6	odd	2		735.2.y.i.263.6		48
15.2	even	4	inner	105.2.x.a.32.7	yes	48
15.8	even	4		525.2.bf.f.32.6		48
15.14	odd	2		525.2.bf.f.368.6		48
21.2	odd	6	inner	105.2.x.a.23.6	yes	48
21.5	even	6		735.2.y.i.128.6		48
21.11	odd	6		735.2.j.g.638.7		24
21.17	even	6		735.2.j.e.638.7		24
21.20	even	2		735.2.y.i.263.7		48
35.2	odd	12	inner	105.2.x.a.2.7	yes	48
35.9	even	6		525.2.bf.f.443.6		48
35.12	even	12		735.2.y.i.422.7		48
35.17	even	12		735.2.j.e.197.7		24
35.23	odd	12		525.2.bf.f.107.6		48
35.27	even	4		735.2.y.i.557.6		48
35.32	odd	12		735.2.j.g.197.7		24
105.2	even	12	inner	105.2.x.a.2.6	✓	48
105.17	odd	12		735.2.j.e.197.6		24
105.23	even	12		525.2.bf.f.107.7		48
105.32	even	12		735.2.j.g.197.6		24
105.44	odd	6		525.2.bf.f.443.7		48
105.47	odd	12		735.2.y.i.422.6		48
105.62	odd	4		735.2.y.i.557.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.6	✓	48	105.2	even	12	inner
105.2.x.a.2.7	yes	48	35.2	odd	12	inner
105.2.x.a.23.6	yes	48	21.2	odd	6	inner
105.2.x.a.23.7	yes	48	7.2	even	3	inner
105.2.x.a.32.6	yes	48	5.2	odd	4	inner
105.2.x.a.32.7	yes	48	15.2	even	4	inner
105.2.x.a.53.6	yes	48	1.1	even	1	trivial
105.2.x.a.53.7	yes	48	3.2	odd	2	inner
525.2.bf.f.32.6		48	15.8	even	4
525.2.bf.f.32.7		48	5.3	odd	4
525.2.bf.f.107.6		48	35.23	odd	12
525.2.bf.f.107.7		48	105.23	even	12
525.2.bf.f.368.6		48	15.14	odd	2
525.2.bf.f.368.7		48	5.4	even	2
525.2.bf.f.443.6		48	35.9	even	6
525.2.bf.f.443.7		48	105.44	odd	6
735.2.j.e.197.6		24	105.17	odd	12
735.2.j.e.197.7		24	35.17	even	12
735.2.j.e.638.6		24	7.3	odd	6
735.2.j.e.638.7		24	21.17	even	6
735.2.j.g.197.6		24	105.32	even	12
735.2.j.g.197.7		24	35.32	odd	12
735.2.j.g.638.6		24	7.4	even	3
735.2.j.g.638.7		24	21.11	odd	6
735.2.y.i.128.6		48	21.5	even	6
735.2.y.i.128.7		48	7.5	odd	6
735.2.y.i.263.6		48	7.6	odd	2
735.2.y.i.263.7		48	21.20	even	2
735.2.y.i.422.6		48	105.47	odd	12
735.2.y.i.422.7		48	35.12	even	12
735.2.y.i.557.6		48	35.27	even	4
735.2.y.i.557.7		48	105.62	odd	4