Newform orbit 525.2.t.a

• Label: 525.2.t.a
• Level: $525$
• Weight: $2$
• Character orbit: 525.t
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 105)
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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 2) q^{2} + (2 \zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - 3 \zeta_{6} q^{6} + ( - \zeta_{6} + 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + q^{4} - 3 q^{6} + 5 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

26.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.50000

−

0.866025i

−

1.73205i

0.500000

+

0.866025i

0

−1.50000

+

2.59808i

2.50000

+

0.866025i

1.73205i

−3.00000

0

101.1

−1.50000

+

0.866025i

1.73205i

0.500000

−

0.866025i

0

−1.50000

−

2.59808i

2.50000

−

0.866025i

−

1.73205i

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.t.a		2
3.b	odd	2	1		525.2.t.e		2
5.b	even	2	1		105.2.s.b	yes	2
5.c	odd	4	2		525.2.q.a		4
7.d	odd	6	1		525.2.t.e		2
15.d	odd	2	1		105.2.s.a	✓	2
15.e	even	4	2		525.2.q.b		4
21.g	even	6	1	inner	525.2.t.a		2
35.c	odd	2	1		735.2.s.e		2
35.i	odd	6	1		105.2.s.a	✓	2
35.i	odd	6	1		735.2.b.b		2
35.j	even	6	1		735.2.b.a		2
35.j	even	6	1		735.2.s.c		2
35.k	even	12	2		525.2.q.b		4
105.g	even	2	1		735.2.s.c		2
105.o	odd	6	1		735.2.b.b		2
105.o	odd	6	1		735.2.s.e		2
105.p	even	6	1		105.2.s.b	yes	2
105.p	even	6	1		735.2.b.a		2
105.w	odd	12	2		525.2.q.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.s.a	✓	2	15.d	odd	2	1
105.2.s.a	✓	2	35.i	odd	6	1
105.2.s.b	yes	2	5.b	even	2	1
105.2.s.b	yes	2	105.p	even	6	1
525.2.q.a		4	5.c	odd	4	2
525.2.q.a		4	105.w	odd	12	2
525.2.q.b		4	15.e	even	4	2
525.2.q.b		4	35.k	even	12	2
525.2.t.a		2	1.a	even	1	1	trivial
525.2.t.a		2	21.g	even	6	1	inner
525.2.t.e		2	3.b	odd	2	1
525.2.t.e		2	7.d	odd	6	1
735.2.b.a		2	35.j	even	6	1
735.2.b.a		2	105.p	even	6	1
735.2.b.b		2	35.i	odd	6	1
735.2.b.b		2	105.o	odd	6	1
735.2.s.c		2	35.j	even	6	1
735.2.s.c		2	105.g	even	2	1
735.2.s.e		2	35.c	odd	2	1
735.2.s.e		2	105.o	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 3T_{2} + 3 \)

\( T_{13}^{2} + 12 \)

\( T_{37}^{2} - 4T_{37} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T + 3 \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 5T + 7 \)
$11$	\( T^{2} - 6T + 12 \)