Newform orbit 189.2.p.a

• Label: 189.2.p.a
• Level: $189$
• Weight: $2$
• Character orbit: 189.p
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$189 = 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	189.p (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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 + (2*z - 2) * q^4 + (3*z - 1) * q^7 + (8*z - 4) * q^13 - 4*z * q^16 + (-3*z + 6) * q^19 + (-5*z + 5) * q^25 + (-2*z - 4) * q^28 + (-z - 1) * q^31 - 10*z * q^37 + 13 * q^43 + (3*z - 8) * q^49 + (-8*z - 8) * q^52 + (5*z - 10) * q^61 + 8 * q^64 + (-16*z + 16) * q^67 + (9*z + 9) * q^73 + (12*z - 6) * q^76 + 4*z * q^79 + (4*z - 20) * q^91 + (22*z - 11) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{4} + q^{7} - 4 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 3 q^{31} - 10 q^{37} + 26 q^{43} - 13 q^{49} - 24 q^{52} - 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} + 4 q^{79} - 36 q^{91}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$\zeta_{6}$

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

0

−1.00000

−

1.73205i

0

0

0.500000

−

2.59808i

0

0

0

80.1

0

0

−1.00000

+

1.73205i

0

0

0.500000

+

2.59808i

0

0

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	189.2.p.a	✓	2
3.b	odd	2	1	CM	189.2.p.a	✓	2
7.c	even	3	1		1323.2.c.a		2
7.d	odd	6	1	inner	189.2.p.a	✓	2
7.d	odd	6	1		1323.2.c.a		2
9.c	even	3	1		567.2.i.a		2
9.c	even	3	1		567.2.s.b		2
9.d	odd	6	1		567.2.i.a		2
9.d	odd	6	1		567.2.s.b		2
21.g	even	6	1	inner	189.2.p.a	✓	2
21.g	even	6	1		1323.2.c.a		2
21.h	odd	6	1		1323.2.c.a		2
63.i	even	6	1		567.2.s.b		2
63.k	odd	6	1		567.2.i.a		2
63.s	even	6	1		567.2.i.a		2
63.t	odd	6	1		567.2.s.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.p.a	✓	2	1.a	even	1	1	trivial
189.2.p.a	✓	2	3.b	odd	2	1	CM
189.2.p.a	✓	2	7.d	odd	6	1	inner
189.2.p.a	✓	2	21.g	even	6	1	inner
567.2.i.a		2	9.c	even	3	1
567.2.i.a		2	9.d	odd	6	1
567.2.i.a		2	63.k	odd	6	1
567.2.i.a		2	63.s	even	6	1
567.2.s.b		2	9.c	even	3	1
567.2.s.b		2	9.d	odd	6	1
567.2.s.b		2	63.i	even	6	1
567.2.s.b		2	63.t	odd	6	1
1323.2.c.a		2	7.c	even	3	1
1323.2.c.a		2	7.d	odd	6	1
1323.2.c.a		2	21.g	even	6	1
1323.2.c.a		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}$ acting on $S_{2}^{\mathrm{new}}(189, [\chi])$.

Hecke characteristic polynomials

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