Embedded newform 525.2.bf.f.368.6

• Label: 525.2.bf.f.368.6
• Level: $525$
• Weight: $2$
• Character: 525.368
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $48$
• 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			368.6
Character	\(\chi\)	\(=\)	525.368
Dual form			525.2.bf.f.107.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0799329 + 0.298314i) q^{2} +(-0.540759 + 1.64547i) q^{3} +(1.64945 + 0.952310i) q^{4} +(-0.447643 - 0.292843i) q^{6} +(2.46856 + 0.951942i) q^{7} +(-0.852694 + 0.852694i) q^{8} +(-2.41516 - 1.77961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{3} - 24 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0799329	+	0.298314i	−0.0565211	+	0.210940i	−0.988411	−	0.151802i	\(-0.951492\pi\)
							0.931890	+	0.362741i	\(0.118159\pi\)
\(3\)	−0.540759	+	1.64547i	−0.312207	+	0.950014i
							\(5\)		0			0
\(7\)	2.46856	+	0.951942i	0.933029	+	0.359800i
							\(11\)	0.660315	+	0.381233i	0.199092	+	0.114946i	0.596232	−	0.802812i	\(-0.296664\pi\)
−0.397140	+	0.917758i	\(0.629997\pi\)
\(13\)	2.27077	+	2.27077i	0.629797	+	0.629797i	0.948017	−	0.318220i	\(-0.103085\pi\)
							−0.318220	+	0.948017i	\(0.603085\pi\)
\(17\)	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	\(-0.651757\pi\)
							−0.458900	+	0.888488i	\(0.651757\pi\)
\(31\)	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	−0.0331401	−	0.999451i	\(-0.510551\pi\)
							−0.528426	+	0.848980i	\(0.677217\pi\)
\(41\)			4.35963i			0.680860i	0.940270	+	0.340430i	\(0.110573\pi\)
							−0.940270	+	0.340430i	\(0.889427\pi\)
\(43\)	−2.69037	−	2.69037i	−0.410277	−	0.410277i	0.471558	−	0.881835i	\(-0.343692\pi\)
							−0.881835	+	0.471558i	\(0.843692\pi\)
\(47\)	−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	525.2.bf.f.368.6		48
3.2	odd	2	inner	525.2.bf.f.368.7		48
5.2	odd	4	inner	525.2.bf.f.32.6		48
5.3	odd	4		105.2.x.a.32.7	yes	48
5.4	even	2		105.2.x.a.53.7	yes	48
7.2	even	3	inner	525.2.bf.f.443.7		48
15.2	even	4	inner	525.2.bf.f.32.7		48
15.8	even	4		105.2.x.a.32.6	yes	48
15.14	odd	2		105.2.x.a.53.6	yes	48
21.2	odd	6	inner	525.2.bf.f.443.6		48
35.2	odd	12	inner	525.2.bf.f.107.7		48
35.3	even	12		735.2.j.e.197.6		24
35.4	even	6		735.2.j.g.638.7		24
35.9	even	6		105.2.x.a.23.6	yes	48
35.13	even	4		735.2.y.i.557.7		48
35.18	odd	12		735.2.j.g.197.6		24
35.19	odd	6		735.2.y.i.128.6		48
35.23	odd	12		105.2.x.a.2.6	✓	48
35.24	odd	6		735.2.j.e.638.7		24
35.33	even	12		735.2.y.i.422.6		48
35.34	odd	2		735.2.y.i.263.7		48
105.2	even	12	inner	525.2.bf.f.107.6		48
105.23	even	12		105.2.x.a.2.7	yes	48
105.38	odd	12		735.2.j.e.197.7		24
105.44	odd	6		105.2.x.a.23.7	yes	48
105.53	even	12		735.2.j.g.197.7		24
105.59	even	6		735.2.j.e.638.6		24
105.68	odd	12		735.2.y.i.422.7		48
105.74	odd	6		735.2.j.g.638.6		24
105.83	odd	4		735.2.y.i.557.6		48
105.89	even	6		735.2.y.i.128.7		48
105.104	even	2		735.2.y.i.263.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.x.a.2.6	✓	48	35.23	odd	12
105.2.x.a.2.7	yes	48	105.23	even	12
105.2.x.a.23.6	yes	48	35.9	even	6
105.2.x.a.23.7	yes	48	105.44	odd	6
105.2.x.a.32.6	yes	48	15.8	even	4
105.2.x.a.32.7	yes	48	5.3	odd	4
105.2.x.a.53.6	yes	48	15.14	odd	2
105.2.x.a.53.7	yes	48	5.4	even	2
525.2.bf.f.32.6		48	5.2	odd	4	inner
525.2.bf.f.32.7		48	15.2	even	4	inner
525.2.bf.f.107.6		48	105.2	even	12	inner
525.2.bf.f.107.7		48	35.2	odd	12	inner
525.2.bf.f.368.6		48	1.1	even	1	trivial
525.2.bf.f.368.7		48	3.2	odd	2	inner
525.2.bf.f.443.6		48	21.2	odd	6	inner
525.2.bf.f.443.7		48	7.2	even	3	inner
735.2.j.e.197.6		24	35.3	even	12
735.2.j.e.197.7		24	105.38	odd	12
735.2.j.e.638.6		24	105.59	even	6
735.2.j.e.638.7		24	35.24	odd	6
735.2.j.g.197.6		24	35.18	odd	12
735.2.j.g.197.7		24	105.53	even	12
735.2.j.g.638.6		24	105.74	odd	6
735.2.j.g.638.7		24	35.4	even	6
735.2.y.i.128.6		48	35.19	odd	6
735.2.y.i.128.7		48	105.89	even	6
735.2.y.i.263.6		48	105.104	even	2
735.2.y.i.263.7		48	35.34	odd	2
735.2.y.i.422.6		48	35.33	even	12
735.2.y.i.422.7		48	105.68	odd	12
735.2.y.i.557.6		48	105.83	odd	4
735.2.y.i.557.7		48	35.13	even	4