Newform orbit 189.2.p.c

• Label: 189.2.p.c
• Level: $189$
• Weight: $2$
• Character orbit: 189.p
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• 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:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 5x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (2 \beta_{2} - 3) q^{7} + \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 8 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 5 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 5 \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{2}\)

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

−1.93649	−	1.11803i
1.93649	+	1.11803i
−1.93649	+	1.11803i
1.93649	−	1.11803i

−1.93649

−

1.11803i

0

1.50000

+

2.59808i

0

0

−2.00000

+

1.73205i

−

2.23607i

0

0

26.2

1.93649

+

1.11803i

0

1.50000

+

2.59808i

0

0

−2.00000

+

1.73205i

2.23607i

0

0

80.1

−1.93649

+

1.11803i

0

1.50000

−

2.59808i

0

0

−2.00000

−

1.73205i

2.23607i

0

0

80.2

1.93649

−

1.11803i

0

1.50000

−

2.59808i

0

0

−2.00000

−

1.73205i

−

2.23607i

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
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.c	✓	4
3.b	odd	2	1	inner	189.2.p.c	✓	4
7.c	even	3	1		1323.2.c.b		4
7.d	odd	6	1	inner	189.2.p.c	✓	4
7.d	odd	6	1		1323.2.c.b		4
9.c	even	3	1		567.2.i.c		4
9.c	even	3	1		567.2.s.e		4
9.d	odd	6	1		567.2.i.c		4
9.d	odd	6	1		567.2.s.e		4
21.g	even	6	1	inner	189.2.p.c	✓	4
21.g	even	6	1		1323.2.c.b		4
21.h	odd	6	1		1323.2.c.b		4
63.i	even	6	1		567.2.s.e		4
63.k	odd	6	1		567.2.i.c		4
63.s	even	6	1		567.2.i.c		4
63.t	odd	6	1		567.2.s.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.p.c	✓	4	1.a	even	1	1	trivial
189.2.p.c	✓	4	3.b	odd	2	1	inner
189.2.p.c	✓	4	7.d	odd	6	1	inner
189.2.p.c	✓	4	21.g	even	6	1	inner
567.2.i.c		4	9.c	even	3	1
567.2.i.c		4	9.d	odd	6	1
567.2.i.c		4	63.k	odd	6	1
567.2.i.c		4	63.s	even	6	1
567.2.s.e		4	9.c	even	3	1
567.2.s.e		4	9.d	odd	6	1
567.2.s.e		4	63.i	even	6	1
567.2.s.e		4	63.t	odd	6	1
1323.2.c.b		4	7.c	even	3	1
1323.2.c.b		4	7.d	odd	6	1
1323.2.c.b		4	21.g	even	6	1
1323.2.c.b		4	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 5T^{2} + 25 \)
$3$	\( T^{4} \)
$5$	\( T^{4} \)
$7$	\( (T^{2} + 4 T + 7)^{2} \)
$11$	\( T^{4} - 20T^{2} + 400 \)