Embedded newform 105.2.x.a.2.4

• Label: 105.2.x.a.2.4
• Level: $105$
• Weight: $2$
• Character: 105.2
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• 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			2.4
Character	\(\chi\)	\(=\)	105.2
Dual form			105.2.x.a.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.340162 - 1.26950i) q^{2} +(-0.664627 + 1.59946i) q^{3} +(0.236127 - 0.136328i) q^{4} +(1.25032 + 1.85383i) q^{5} +(2.25660 + 0.299670i) q^{6} +(2.32676 + 1.25943i) q^{7} +(-2.11207 - 2.11207i) q^{8} +(-2.11654 - 2.12609i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.340162	−	1.26950i	−0.240531	−	0.897673i	−0.975577	−	0.219656i	\(-0.929506\pi\)
							0.735046	−	0.678017i	\(-0.237160\pi\)
\(3\)	−0.664627	+	1.59946i	−0.383723	+	0.923448i
							\(5\)	1.25032	+	1.85383i	0.559161	+	0.829059i
\(7\)	2.32676	+	1.25943i	0.879434	+	0.476021i
							\(11\)	3.38224	−	1.95274i	1.01978	−	0.588773i	0.105743	−	0.994394i	\(-0.466278\pi\)
0.914041	+	0.405621i	\(0.132945\pi\)
\(13\)	−1.56642	+	1.56642i	−0.434448	+	0.434448i	−0.890138	−	0.455691i	\(-0.849392\pi\)
							0.455691	+	0.890138i	\(0.349392\pi\)
\(17\)	−2.58656	−	0.693065i	−0.627332	−	0.168093i	−0.0688731	−	0.997625i	\(-0.521940\pi\)
							−0.558459	+	0.829532i	\(0.688607\pi\)
\(19\)	−1.61097	−	0.930096i	−0.369582	−	0.213379i	0.303694	−	0.952770i	\(-0.401780\pi\)
							−0.673276	+	0.739391i	\(0.735113\pi\)
\(23\)	−2.38315	+	0.638564i	−0.496921	+	0.133150i	−0.498571	−	0.866849i	\(-0.666142\pi\)
							0.00164943	+	0.999999i	\(0.499475\pi\)
\(29\)	0.513153			0.0952901			0.0476450	−	0.998864i	\(-0.484828\pi\)
							0.0476450	+	0.998864i	\(0.484828\pi\)
\(31\)	−4.29138	−	7.43289i	−0.770755	−	1.33499i	−0.937150	−	0.348928i	\(-0.886546\pi\)
							0.166394	−	0.986059i	\(-0.446788\pi\)
\(37\)	−6.60698	+	1.77034i	−1.08618	+	0.291041i	−0.757126	−	0.653269i	\(-0.773397\pi\)
							−0.329056	+	0.944311i	\(0.606730\pi\)
\(41\)			0.308469i			0.0481748i	0.999710	+	0.0240874i	\(0.00766800\pi\)
							−0.999710	+	0.0240874i	\(0.992332\pi\)
\(43\)	7.60892	−	7.60892i	1.16035	−	1.16035i	0.175950	−	0.984399i	\(-0.443700\pi\)
							0.984399	−	0.175950i	\(-0.0562999\pi\)
\(47\)	−1.36920	−	5.10994i	−0.199719	−	0.745361i	−0.990995	−	0.133902i	\(-0.957249\pi\)
							0.791276	−	0.611460i	\(-0.209417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.x.a.2.4	✓	48
3.2	odd	2	inner	105.2.x.a.2.9	yes	48
5.2	odd	4		525.2.bf.f.443.9		48
5.3	odd	4	inner	105.2.x.a.23.4	yes	48
5.4	even	2		525.2.bf.f.107.9		48
7.2	even	3		735.2.j.g.197.4		24
7.3	odd	6		735.2.y.i.557.9		48
7.4	even	3	inner	105.2.x.a.32.9	yes	48
7.5	odd	6		735.2.j.e.197.4		24
7.6	odd	2		735.2.y.i.422.4		48
15.2	even	4		525.2.bf.f.443.4		48
15.8	even	4	inner	105.2.x.a.23.9	yes	48
15.14	odd	2		525.2.bf.f.107.4		48
21.2	odd	6		735.2.j.g.197.9		24
21.5	even	6		735.2.j.e.197.9		24
21.11	odd	6	inner	105.2.x.a.32.4	yes	48
21.17	even	6		735.2.y.i.557.4		48
21.20	even	2		735.2.y.i.422.9		48
35.3	even	12		735.2.y.i.263.9		48
35.4	even	6		525.2.bf.f.32.4		48
35.13	even	4		735.2.y.i.128.4		48
35.18	odd	12	inner	105.2.x.a.53.9	yes	48
35.23	odd	12		735.2.j.g.638.9		24
35.32	odd	12		525.2.bf.f.368.4		48
35.33	even	12		735.2.j.e.638.9		24
105.23	even	12		735.2.j.g.638.4		24
105.32	even	12		525.2.bf.f.368.9		48
105.38	odd	12		735.2.y.i.263.4		48
105.53	even	12	inner	105.2.x.a.53.4	yes	48
105.68	odd	12		735.2.j.e.638.4		24
105.74	odd	6		525.2.bf.f.32.9		48
105.83	odd	4		735.2.y.i.128.9		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.4	✓	48	1.1	even	1	trivial
105.2.x.a.2.9	yes	48	3.2	odd	2	inner
105.2.x.a.23.4	yes	48	5.3	odd	4	inner
105.2.x.a.23.9	yes	48	15.8	even	4	inner
105.2.x.a.32.4	yes	48	21.11	odd	6	inner
105.2.x.a.32.9	yes	48	7.4	even	3	inner
105.2.x.a.53.4	yes	48	105.53	even	12	inner
105.2.x.a.53.9	yes	48	35.18	odd	12	inner
525.2.bf.f.32.4		48	35.4	even	6
525.2.bf.f.32.9		48	105.74	odd	6
525.2.bf.f.107.4		48	15.14	odd	2
525.2.bf.f.107.9		48	5.4	even	2
525.2.bf.f.368.4		48	35.32	odd	12
525.2.bf.f.368.9		48	105.32	even	12
525.2.bf.f.443.4		48	15.2	even	4
525.2.bf.f.443.9		48	5.2	odd	4
735.2.j.e.197.4		24	7.5	odd	6
735.2.j.e.197.9		24	21.5	even	6
735.2.j.e.638.4		24	105.68	odd	12
735.2.j.e.638.9		24	35.33	even	12
735.2.j.g.197.4		24	7.2	even	3
735.2.j.g.197.9		24	21.2	odd	6
735.2.j.g.638.4		24	105.23	even	12
735.2.j.g.638.9		24	35.23	odd	12
735.2.y.i.128.4		48	35.13	even	4
735.2.y.i.128.9		48	105.83	odd	4
735.2.y.i.263.4		48	105.38	odd	12
735.2.y.i.263.9		48	35.3	even	12
735.2.y.i.422.4		48	7.6	odd	2
735.2.y.i.422.9		48	21.20	even	2
735.2.y.i.557.4		48	21.17	even	6
735.2.y.i.557.9		48	7.3	odd	6