Newform orbit 525.3.h.a

• Label: 525.3.h.a
• Level: $525$
• Weight: $3$
• Character orbit: 525.h
• 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.h (of order \(2\), degree \(1\), 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:	\( 2 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + \beta q^{3} - 3 q^{4} - \beta q^{6} + (4 \beta - 1) q^{7} + 7 q^{8} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{4} - 2 q^{7} + 14 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

76.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−3.00000

0

1.73205i

−1.00000

−

6.92820i

7.00000

−3.00000

0

76.2

−1.00000

1.73205i

−3.00000

0

−

1.73205i

−1.00000

+

6.92820i

7.00000

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.3.h.a		2
5.b	even	2	1		21.3.d.a	✓	2
5.c	odd	4	2		525.3.e.a		4
7.b	odd	2	1	inner	525.3.h.a		2
15.d	odd	2	1		63.3.d.b		2
20.d	odd	2	1		336.3.f.a		2
35.c	odd	2	1		21.3.d.a	✓	2
35.f	even	4	2		525.3.e.a		4
35.i	odd	6	1		147.3.f.b		2
35.i	odd	6	1		147.3.f.d		2
35.j	even	6	1		147.3.f.b		2
35.j	even	6	1		147.3.f.d		2
40.e	odd	2	1		1344.3.f.b		2
40.f	even	2	1		1344.3.f.c		2
60.h	even	2	1		1008.3.f.d		2
105.g	even	2	1		63.3.d.b		2
105.o	odd	6	1		441.3.m.d		2
105.o	odd	6	1		441.3.m.f		2
105.p	even	6	1		441.3.m.d		2
105.p	even	6	1		441.3.m.f		2
140.c	even	2	1		336.3.f.a		2
280.c	odd	2	1		1344.3.f.c		2
280.n	even	2	1		1344.3.f.b		2
420.o	odd	2	1		1008.3.f.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.3.d.a	✓	2	5.b	even	2	1
21.3.d.a	✓	2	35.c	odd	2	1
63.3.d.b		2	15.d	odd	2	1
63.3.d.b		2	105.g	even	2	1
147.3.f.b		2	35.i	odd	6	1
147.3.f.b		2	35.j	even	6	1
147.3.f.d		2	35.i	odd	6	1
147.3.f.d		2	35.j	even	6	1
336.3.f.a		2	20.d	odd	2	1
336.3.f.a		2	140.c	even	2	1
441.3.m.d		2	105.o	odd	6	1
441.3.m.d		2	105.p	even	6	1
441.3.m.f		2	105.o	odd	6	1
441.3.m.f		2	105.p	even	6	1
525.3.e.a		4	5.c	odd	4	2
525.3.e.a		4	35.f	even	4	2
525.3.h.a		2	1.a	even	1	1	trivial
525.3.h.a		2	7.b	odd	2	1	inner
1008.3.f.d		2	60.h	even	2	1
1008.3.f.d		2	420.o	odd	2	1
1344.3.f.b		2	40.e	odd	2	1
1344.3.f.b		2	280.n	even	2	1
1344.3.f.c		2	40.f	even	2	1
1344.3.f.c		2	280.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 2T + 49 \)
$11$	\( (T - 10)^{2} \)