Newform orbit 525.2.t.b

• Label: 525.2.t.b
• Level: $525$
• Weight: $2$
• Character orbit: 525.t
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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 + (-z - 1) * q^3 + (2*z - 2) * q^4 + (2*z - 3) * q^7 + 3*z * q^9 + (-2*z + 4) * q^12 + (-8*z + 4) * q^13 - 4*z * q^16 + (-2*z + 4) * q^19 + (-z + 5) * q^21 + (-6*z + 3) * q^27 + (-6*z + 2) * q^28 + (-5*z - 5) * q^31 - 6 * q^36 - 11*z * q^37 + (12*z - 12) * q^39 + 5 * q^43 + (8*z - 4) * q^48 + (-8*z + 5) * q^49 + (8*z + 8) * q^52 - 6 * q^57 + (-9*z + 18) * q^61 + (-3*z - 6) * q^63 + 8 * q^64 + (16*z - 16) * q^67 + (z + 1) * q^73 + (8*z - 4) * q^76 - 17*z * q^79 + (9*z - 9) * q^81 + (10*z - 8) * q^84 + (16*z + 4) * q^91 + 15*z * q^93 + (6*z - 3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 3 * q^3 - 2 * q^4 - 4 * q^7 + 3 * q^9 + 6 * q^12 - 4 * q^16 + 6 * q^19 + 9 * q^21 - 2 * q^28 - 15 * q^31 - 12 * q^36 - 11 * q^37 - 12 * q^39 + 10 * q^43 + 2 * q^49 + 24 * q^52 - 12 * q^57 + 27 * q^61 - 15 * q^63 + 16 * q^64 - 16 * q^67 + 3 * q^73 - 17 * q^79 - 9 * q^81 - 6 * q^84 + 24 * q^91 + 15 * q^93

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

0

−1.50000

+

0.866025i

−1.00000

−

1.73205i

0

0

−2.00000

−

1.73205i

0

1.50000

−

2.59808i

0

101.1

0

−1.50000

−

0.866025i

−1.00000

+

1.73205i

0

0

−2.00000

+

1.73205i

0

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
7.d	odd	6	1	inner
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.b	✓	2
3.b	odd	2	1	CM	525.2.t.b	✓	2
5.b	even	2	1		525.2.t.d	yes	2
5.c	odd	4	2		525.2.q.c		4
7.d	odd	6	1	inner	525.2.t.b	✓	2
15.d	odd	2	1		525.2.t.d	yes	2
15.e	even	4	2		525.2.q.c		4
21.g	even	6	1	inner	525.2.t.b	✓	2
35.i	odd	6	1		525.2.t.d	yes	2
35.k	even	12	2		525.2.q.c		4
105.p	even	6	1		525.2.t.d	yes	2
105.w	odd	12	2		525.2.q.c		4

    

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

Hecke kernels

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

$T_{2}$

$T_{13}^{2} + 48$

$T_{37}^{2} + 11T_{37} + 121$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 3T + 3$
$5$	$T^{2}$
$7$	$T^{2} + 4T + 7$
$11$	$T^{2}$