Newform orbit 189.2.c.b

• Label: 189.2.c.b
• Level: $189$
• Weight: $2$
• Character orbit: 189.c
• 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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 5x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} - 3 q^{4} - \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_1 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 8 q^{7}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} ) / 5 \)
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{2} - 5 ) / 5 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 10\nu ) / 5 \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

188.1

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

−

2.23607i

0

−3.00000

−3.87298

0

−2.00000

+

1.73205i

2.23607i

0

8.66025i

188.2

−

2.23607i

0

−3.00000

3.87298

0

−2.00000

−

1.73205i

2.23607i

0

−

8.66025i

188.3

2.23607i

0

−3.00000

−3.87298

0

−2.00000

−

1.73205i

−

2.23607i

0

−

8.66025i

188.4

2.23607i

0

−3.00000

3.87298

0

−2.00000

+

1.73205i

−

2.23607i

0

8.66025i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.2.c.b	✓	4
3.b	odd	2	1	inner	189.2.c.b	✓	4
4.b	odd	2	1		3024.2.k.i		4
7.b	odd	2	1	inner	189.2.c.b	✓	4
9.c	even	3	1		567.2.o.c		4
9.c	even	3	1		567.2.o.d		4
9.d	odd	6	1		567.2.o.c		4
9.d	odd	6	1		567.2.o.d		4
12.b	even	2	1		3024.2.k.i		4
21.c	even	2	1	inner	189.2.c.b	✓	4
28.d	even	2	1		3024.2.k.i		4
63.l	odd	6	1		567.2.o.c		4
63.l	odd	6	1		567.2.o.d		4
63.o	even	6	1		567.2.o.c		4
63.o	even	6	1		567.2.o.d		4
84.h	odd	2	1		3024.2.k.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.c.b	✓	4	1.a	even	1	1	trivial
189.2.c.b	✓	4	3.b	odd	2	1	inner
189.2.c.b	✓	4	7.b	odd	2	1	inner
189.2.c.b	✓	4	21.c	even	2	1	inner
567.2.o.c		4	9.c	even	3	1
567.2.o.c		4	9.d	odd	6	1
567.2.o.c		4	63.l	odd	6	1
567.2.o.c		4	63.o	even	6	1
567.2.o.d		4	9.c	even	3	1
567.2.o.d		4	9.d	odd	6	1
567.2.o.d		4	63.l	odd	6	1
567.2.o.d		4	63.o	even	6	1
3024.2.k.i		4	4.b	odd	2	1
3024.2.k.i		4	12.b	even	2	1
3024.2.k.i		4	28.d	even	2	1
3024.2.k.i		4	84.h	odd	2	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 5)^{2} \)
$3$	\( T^{4} \)
$5$	\( (T^{2} - 15)^{2} \)
$7$	\( (T^{2} + 4 T + 7)^{2} \)
$11$	\( (T^{2} + 5)^{2} \)