Embedded newform 525.2.bf.f.32.10

• Label: 525.2.bf.f.32.10
• Level: $525$
• Weight: $2$
• Character: 525.32
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			32.10
Character	\(\chi\)	\(=\)	525.32
Dual form			525.2.bf.f.443.10

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46015	+	0.391246i	1.03248	+	0.276653i	0.734995	−	0.678072i	\(-0.237184\pi\)
							0.297488	+	0.954726i	\(0.403851\pi\)
\(3\)	−1.49243	−	0.879005i	−0.861655	−	0.507494i
							\(5\)		0			0
\(7\)	−2.36949	+	1.17707i	−0.895585	+	0.444891i
							\(11\)	−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	\(-0.956422\pi\)
							0.136477	+	0.990643i	\(0.456422\pi\)
\(17\)	−0.311437	−	1.16230i	−0.0755345	−	0.281899i	0.917819	−	0.396998i	\(-0.129948\pi\)
							−0.993354	+	0.115099i	\(0.963281\pi\)
\(19\)	−5.95337	+	3.43718i	−1.36580	+	0.788543i	−0.990388	−	0.138316i	\(-0.955831\pi\)
							−0.375409	+	0.926859i	\(0.622498\pi\)
\(23\)	0.505926	−	1.88814i	0.105493	−	0.393705i	−0.892908	−	0.450240i	\(-0.851339\pi\)
							0.998401	+	0.0565348i	\(0.0180052\pi\)
\(29\)	−2.72261			−0.505576			−0.252788	−	0.967522i	\(-0.581348\pi\)
							−0.252788	+	0.967522i	\(0.581348\pi\)
\(31\)	−2.31688	+	4.01295i	−0.416123	+	0.720747i	−0.995546	−	0.0942806i	\(-0.969945\pi\)
							0.579422	+	0.815028i	\(0.303278\pi\)
\(37\)	−0.207656	+	0.774982i	−0.0341384	+	0.127406i	−0.980892	−	0.194552i	\(-0.937675\pi\)
							0.946754	+	0.321959i	\(0.104341\pi\)
\(41\)		−	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	\(-0.576468\pi\)
							0.971283	+	0.237928i	\(0.0764681\pi\)
\(47\)	−10.1240	−	2.71272i	−1.47674	−	0.395691i	−0.571503	−	0.820600i	\(-0.693640\pi\)
							−0.905236	+	0.424909i	\(0.860306\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.3	✓	48	105.44	odd	6
105.2.x.a.2.10	yes	48	35.9	even	6
105.2.x.a.23.3	yes	48	105.2	even	12
105.2.x.a.23.10	yes	48	35.2	odd	12
105.2.x.a.32.3	yes	48	5.4	even	2
105.2.x.a.32.10	yes	48	15.14	odd	2
105.2.x.a.53.3	yes	48	5.2	odd	4
105.2.x.a.53.10	yes	48	15.2	even	4
525.2.bf.f.32.3		48	3.2	odd	2	inner
525.2.bf.f.32.10		48	1.1	even	1	trivial
525.2.bf.f.107.3		48	7.2	even	3	inner
525.2.bf.f.107.10		48	21.2	odd	6	inner
525.2.bf.f.368.3		48	15.8	even	4	inner
525.2.bf.f.368.10		48	5.3	odd	4	inner
525.2.bf.f.443.3		48	35.23	odd	12	inner
525.2.bf.f.443.10		48	105.23	even	12	inner
735.2.j.e.197.3		24	105.59	even	6
735.2.j.e.197.10		24	35.24	odd	6
735.2.j.e.638.3		24	35.17	even	12
735.2.j.e.638.10		24	105.17	odd	12
735.2.j.g.197.3		24	105.74	odd	6
735.2.j.g.197.10		24	35.4	even	6
735.2.j.g.638.3		24	35.32	odd	12
735.2.j.g.638.10		24	105.32	even	12
735.2.y.i.128.3		48	105.47	odd	12
735.2.y.i.128.10		48	35.12	even	12
735.2.y.i.263.3		48	35.27	even	4
735.2.y.i.263.10		48	105.62	odd	4
735.2.y.i.422.3		48	105.89	even	6
735.2.y.i.422.10		48	35.19	odd	6
735.2.y.i.557.3		48	35.34	odd	2
735.2.y.i.557.10		48	105.104	even	2