Newform orbit 525.3.o.b

• Label: 525.3.o.b
• Level: $525$
• Weight: $3$
• Character orbit: 525.o
• Analytic conductor: $14.305$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	525.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.3052138789\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 \zeta_{6} q^{2} + (\zeta_{6} + 1) q^{3} + ( - 4 \zeta_{6} + 2) q^{6} + 7 \zeta_{6} q^{7} - 8 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 3 q^{3} + 7 q^{7} - 16 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

376.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

+

1.73205i

1.50000

−

0.866025i

0

0

3.46410i

3.50000

−

6.06218i

−8.00000

1.50000

−

2.59808i

0

451.1

−1.00000

−

1.73205i

1.50000

+

0.866025i

0

0

−

3.46410i

3.50000

+

6.06218i

−8.00000

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.3.o.b		2
5.b	even	2	1		21.3.f.c	✓	2
5.c	odd	4	2		525.3.s.d		4
7.d	odd	6	1	inner	525.3.o.b		2
15.d	odd	2	1		63.3.m.a		2
20.d	odd	2	1		336.3.bh.c		2
35.c	odd	2	1		147.3.f.e		2
35.i	odd	6	1		21.3.f.c	✓	2
35.i	odd	6	1		147.3.d.a		2
35.j	even	6	1		147.3.d.a		2
35.j	even	6	1		147.3.f.e		2
35.k	even	12	2		525.3.s.d		4
60.h	even	2	1		1008.3.cg.f		2
105.g	even	2	1		441.3.m.b		2
105.o	odd	6	1		441.3.d.d		2
105.o	odd	6	1		441.3.m.b		2
105.p	even	6	1		63.3.m.a		2
105.p	even	6	1		441.3.d.d		2
140.p	odd	6	1		2352.3.f.b		2
140.s	even	6	1		336.3.bh.c		2
140.s	even	6	1		2352.3.f.b		2
420.be	odd	6	1		1008.3.cg.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.3.f.c	✓	2	5.b	even	2	1
21.3.f.c	✓	2	35.i	odd	6	1
63.3.m.a		2	15.d	odd	2	1
63.3.m.a		2	105.p	even	6	1
147.3.d.a		2	35.i	odd	6	1
147.3.d.a		2	35.j	even	6	1
147.3.f.e		2	35.c	odd	2	1
147.3.f.e		2	35.j	even	6	1
336.3.bh.c		2	20.d	odd	2	1
336.3.bh.c		2	140.s	even	6	1
441.3.d.d		2	105.o	odd	6	1
441.3.d.d		2	105.p	even	6	1
441.3.m.b		2	105.g	even	2	1
441.3.m.b		2	105.o	odd	6	1
525.3.o.b		2	1.a	even	1	1	trivial
525.3.o.b		2	7.d	odd	6	1	inner
525.3.s.d		4	5.c	odd	4	2
525.3.s.d		4	35.k	even	12	2
1008.3.cg.f		2	60.h	even	2	1
1008.3.cg.f		2	420.be	odd	6	1
2352.3.f.b		2	140.p	odd	6	1
2352.3.f.b		2	140.s	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 2T_{2} + 4 \)

\( T_{11}^{2} + 10T_{11} + 100 \)

\( T_{13}^{2} + 147 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} - 3T + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 7T + 49 \)
$11$	\( T^{2} + 10T + 100 \)