Embedded newform 105.2.x.a.53.3

• Label: 105.2.x.a.53.3
• Level: $105$
• Weight: $2$
• Character: 105.53
• 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			53.3
Character	\(\chi\)	\(=\)	105.53
Dual form			105.2.x.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.391246 + 1.46015i) q^{2} +(-0.879005 + 1.49243i) q^{3} +(-0.246919 - 0.142558i) q^{4} +(1.82416 + 1.29322i) q^{5} +(-1.83527 - 1.86739i) q^{6} +(-1.17707 - 2.36949i) q^{7} +(-1.83305 + 1.83305i) q^{8} +(-1.45470 - 2.62371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 24 q^{6} - 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.391246	+	1.46015i	−0.276653	+	1.03248i	0.678072	+	0.734995i	\(0.262816\pi\)
							−0.954726	+	0.297488i	\(0.903851\pi\)
\(3\)	−0.879005	+	1.49243i	−0.507494	+	0.861655i
							\(5\)	1.82416	+	1.29322i	0.815791	+	0.578347i
\(7\)	−1.17707	−	2.36949i	−0.444891	−	0.895585i
							\(11\)	−0.791646	−	0.457057i	−0.238690	−	0.137808i	0.375884	−	0.926667i	\(-0.377339\pi\)
−0.614575	+	0.788859i	\(0.710672\pi\)
\(13\)	3.07974	+	3.07974i	0.854166	+	0.854166i	0.990643	−	0.136477i	\(-0.0435781\pi\)
							−0.136477	+	0.990643i	\(0.543578\pi\)
\(17\)	1.16230	−	0.311437i	0.281899	−	0.0755345i	−0.115099	−	0.993354i	\(-0.536719\pi\)
							0.396998	+	0.917819i	\(0.370052\pi\)
\(19\)	5.95337	−	3.43718i	1.36580	−	0.788543i	0.375409	−	0.926859i	\(-0.377502\pi\)
							0.990388	+	0.138316i	\(0.0441690\pi\)
\(23\)	−1.88814	−	0.505926i	−0.393705	−	0.105493i	0.0565348	−	0.998401i	\(-0.481995\pi\)
							−0.450240	+	0.892908i	\(0.648661\pi\)
\(29\)	2.72261			0.505576			0.252788	−	0.967522i	\(-0.418652\pi\)
							0.252788	+	0.967522i	\(0.418652\pi\)
\(31\)	−2.31688	+	4.01295i	−0.416123	+	0.720747i	−0.995546	−	0.0942806i	\(-0.969945\pi\)
							0.579422	+	0.815028i	\(0.303278\pi\)
\(37\)	−0.774982	−	0.207656i	−0.127406	−	0.0341384i	0.194552	−	0.980892i	\(-0.437675\pi\)
							−0.321959	+	0.946754i	\(0.604341\pi\)
\(41\)		−	0.922837i		−	0.144123i	−0.997400	−	0.0720615i	\(-0.977042\pi\)
							0.997400	−	0.0720615i	\(-0.0229578\pi\)
\(43\)	−4.80893	−	4.80893i	−0.733355	−	0.733355i	0.237928	−	0.971283i	\(-0.423532\pi\)
							−0.971283	+	0.237928i	\(0.923532\pi\)
\(47\)	2.71272	−	10.1240i	0.395691	−	1.47674i	−0.424909	−	0.905236i	\(-0.639694\pi\)
							0.820600	−	0.571503i	\(-0.193640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.x.a.53.3	yes	48
3.2	odd	2	inner	105.2.x.a.53.10	yes	48
5.2	odd	4	inner	105.2.x.a.32.3	yes	48
5.3	odd	4		525.2.bf.f.32.10		48
5.4	even	2		525.2.bf.f.368.10		48
7.2	even	3	inner	105.2.x.a.23.10	yes	48
7.3	odd	6		735.2.j.e.638.3		24
7.4	even	3		735.2.j.g.638.3		24
7.5	odd	6		735.2.y.i.128.10		48
7.6	odd	2		735.2.y.i.263.3		48
15.2	even	4	inner	105.2.x.a.32.10	yes	48
15.8	even	4		525.2.bf.f.32.3		48
15.14	odd	2		525.2.bf.f.368.3		48
21.2	odd	6	inner	105.2.x.a.23.3	yes	48
21.5	even	6		735.2.y.i.128.3		48
21.11	odd	6		735.2.j.g.638.10		24
21.17	even	6		735.2.j.e.638.10		24
21.20	even	2		735.2.y.i.263.10		48
35.2	odd	12	inner	105.2.x.a.2.10	yes	48
35.9	even	6		525.2.bf.f.443.3		48
35.12	even	12		735.2.y.i.422.10		48
35.17	even	12		735.2.j.e.197.10		24
35.23	odd	12		525.2.bf.f.107.3		48
35.27	even	4		735.2.y.i.557.3		48
35.32	odd	12		735.2.j.g.197.10		24
105.2	even	12	inner	105.2.x.a.2.3	✓	48
105.17	odd	12		735.2.j.e.197.3		24
105.23	even	12		525.2.bf.f.107.10		48
105.32	even	12		735.2.j.g.197.3		24
105.44	odd	6		525.2.bf.f.443.10		48
105.47	odd	12		735.2.y.i.422.3		48
105.62	odd	4		735.2.y.i.557.10		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.3	✓	48	105.2	even	12	inner
105.2.x.a.2.10	yes	48	35.2	odd	12	inner
105.2.x.a.23.3	yes	48	21.2	odd	6	inner
105.2.x.a.23.10	yes	48	7.2	even	3	inner
105.2.x.a.32.3	yes	48	5.2	odd	4	inner
105.2.x.a.32.10	yes	48	15.2	even	4	inner
105.2.x.a.53.3	yes	48	1.1	even	1	trivial
105.2.x.a.53.10	yes	48	3.2	odd	2	inner
525.2.bf.f.32.3		48	15.8	even	4
525.2.bf.f.32.10		48	5.3	odd	4
525.2.bf.f.107.3		48	35.23	odd	12
525.2.bf.f.107.10		48	105.23	even	12
525.2.bf.f.368.3		48	15.14	odd	2
525.2.bf.f.368.10		48	5.4	even	2
525.2.bf.f.443.3		48	35.9	even	6
525.2.bf.f.443.10		48	105.44	odd	6
735.2.j.e.197.3		24	105.17	odd	12
735.2.j.e.197.10		24	35.17	even	12
735.2.j.e.638.3		24	7.3	odd	6
735.2.j.e.638.10		24	21.17	even	6
735.2.j.g.197.3		24	105.32	even	12
735.2.j.g.197.10		24	35.32	odd	12
735.2.j.g.638.3		24	7.4	even	3
735.2.j.g.638.10		24	21.11	odd	6
735.2.y.i.128.3		48	21.5	even	6
735.2.y.i.128.10		48	7.5	odd	6
735.2.y.i.263.3		48	7.6	odd	2
735.2.y.i.263.10		48	21.20	even	2
735.2.y.i.422.3		48	105.47	odd	12
735.2.y.i.422.10		48	35.12	even	12
735.2.y.i.557.3		48	35.27	even	4
735.2.y.i.557.10		48	105.62	odd	4