Embedded newform 525.2.bc.e.82.7

• Label: 525.2.bc.e.82.7
• Level: $525$
• Weight: $2$
• Character: 525.82
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			82.7
Character	\(\chi\)	\(=\)	525.82
Dual form			525.2.bc.e.493.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.24814 - 0.602389i) q^{2} +(-0.258819 + 0.965926i) q^{3} +(2.95923 - 1.70851i) q^{4} +2.32745i q^{6} +(2.59417 - 0.519864i) q^{7} +(2.33208 - 2.33208i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.24814	−	0.602389i	1.58968	−	0.425953i	0.647774	−	0.761833i	\(-0.275700\pi\)
							0.941905	+	0.335880i	\(0.109034\pi\)
\(3\)	−0.258819	+	0.965926i	−0.149429	+	0.557678i
							\(5\)		0			0
\(7\)	2.59417	−	0.519864i	0.980506	−	0.196490i
							\(11\)	−1.76389	−	3.05515i	−0.531834	−	0.921163i	−0.999309	−	0.0371569i	\(-0.988170\pi\)
0.467476	−	0.884006i	\(-0.345163\pi\)
\(13\)	4.49057	+	4.49057i	1.24546	+	1.24546i	0.957704	+	0.287756i	\(0.0929093\pi\)
							0.287756	+	0.957704i	\(0.407091\pi\)
\(17\)	−1.79795	−	0.481759i	−0.436066	−	0.116844i	0.0341060	−	0.999418i	\(-0.489142\pi\)
							−0.470172	+	0.882575i	\(0.655808\pi\)
\(19\)	0.0699116	−	0.121090i	0.0160388	−	0.0277800i	−0.857895	−	0.513826i	\(-0.828228\pi\)
							0.873933	+	0.486046i	\(0.161561\pi\)
\(23\)	−0.997072	−	3.72112i	−0.207904	−	0.775908i	−0.988545	−	0.150928i	\(-0.951774\pi\)
							0.780641	−	0.624980i	\(-0.214893\pi\)
\(29\)		−	2.01969i		−	0.375047i	−0.982260	−	0.187524i	\(-0.939954\pi\)
							0.982260	−	0.187524i	\(-0.0600462\pi\)
\(31\)	−4.56612	+	2.63625i	−0.820100	+	0.473485i	−0.850451	−	0.526054i	\(-0.823671\pi\)
							0.0303510	+	0.999539i	\(0.490337\pi\)
\(37\)	−5.61323	+	1.50406i	−0.922810	+	0.247266i	−0.688786	−	0.724965i	\(-0.741856\pi\)
							−0.234024	+	0.972231i	\(0.575189\pi\)
\(41\)		−	0.903323i		−	0.141075i	−0.997509	−	0.0705377i	\(-0.977529\pi\)
							0.997509	−	0.0705377i	\(-0.0224715\pi\)
\(43\)	−2.38469	+	2.38469i	−0.363663	+	0.363663i	−0.865159	−	0.501497i	\(-0.832783\pi\)
							0.501497	+	0.865159i	\(0.332783\pi\)
\(47\)	0.639474	+	2.38655i	0.0932768	+	0.348114i	0.996753	−	0.0805254i	\(-0.0256598\pi\)
							−0.903476	+	0.428639i	\(0.858993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bc.e.82.7		32
5.2	odd	4		105.2.u.a.103.7	yes	32
5.3	odd	4	inner	525.2.bc.e.418.2		32
5.4	even	2		105.2.u.a.82.2	yes	32
7.3	odd	6	inner	525.2.bc.e.157.2		32
15.2	even	4		315.2.bz.d.208.2		32
15.14	odd	2		315.2.bz.d.82.7		32
35.2	odd	12		735.2.m.c.538.13		32
35.3	even	12	inner	525.2.bc.e.493.7		32
35.4	even	6		735.2.v.b.472.7		32
35.9	even	6		735.2.m.c.97.14		32
35.12	even	12		735.2.m.c.538.14		32
35.17	even	12		105.2.u.a.73.2	yes	32
35.19	odd	6		735.2.m.c.97.13		32
35.24	odd	6		105.2.u.a.52.7	✓	32
35.27	even	4		735.2.v.b.313.7		32
35.32	odd	12		735.2.v.b.178.2		32
35.34	odd	2		735.2.v.b.607.2		32
105.17	odd	12		315.2.bz.d.73.7		32
105.59	even	6		315.2.bz.d.262.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.7	✓	32	35.24	odd	6
105.2.u.a.73.2	yes	32	35.17	even	12
105.2.u.a.82.2	yes	32	5.4	even	2
105.2.u.a.103.7	yes	32	5.2	odd	4
315.2.bz.d.73.7		32	105.17	odd	12
315.2.bz.d.82.7		32	15.14	odd	2
315.2.bz.d.208.2		32	15.2	even	4
315.2.bz.d.262.2		32	105.59	even	6
525.2.bc.e.82.7		32	1.1	even	1	trivial
525.2.bc.e.157.2		32	7.3	odd	6	inner
525.2.bc.e.418.2		32	5.3	odd	4	inner
525.2.bc.e.493.7		32	35.3	even	12	inner
735.2.m.c.97.13		32	35.19	odd	6
735.2.m.c.97.14		32	35.9	even	6
735.2.m.c.538.13		32	35.2	odd	12
735.2.m.c.538.14		32	35.12	even	12
735.2.v.b.178.2		32	35.32	odd	12
735.2.v.b.313.7		32	35.27	even	4
735.2.v.b.472.7		32	35.4	even	6
735.2.v.b.607.2		32	35.34	odd	2