Newform orbit 189.2.f.a

• Label: 189.2.f.a
• Level: $189$
• Weight: $2$
• Character orbit: 189.f
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.309123.1
Defining polynomial:	\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{4} + (2 \beta_{4} + \beta_{2} - 2) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_1 + 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 3 q^{4} - 5 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) :

\(\beta_{1}\)	\(=\)	\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \)
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(-1 + \beta_{4}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

64.1

0.500000	−	2.05195i
0.500000	+	1.41036i
0.500000	−	0.224437i
0.500000	+	2.05195i
0.500000	−	1.41036i
0.500000	+	0.224437i

−1.23025

−

2.13086i

0

−2.02704

+

3.51094i

−1.29679

+

2.24611i

0

0.500000

+

0.866025i

5.05408

0

6.38151

64.2

−0.119562

−

0.207087i

0

0.971410

−

1.68253i

0.590972

−

1.02359i

0

0.500000

+

0.866025i

−0.942820

0

−0.282630

64.3

0.849814

+

1.47192i

0

−0.444368

+

0.769668i

−1.79418

+

3.10761i

0

0.500000

+

0.866025i

1.88874

0

−6.09888

127.1

−1.23025

+

2.13086i

0

−2.02704

−

3.51094i

−1.29679

−

2.24611i

0

0.500000

−

0.866025i

5.05408

0

6.38151

127.2

−0.119562

+

0.207087i

0

0.971410

+

1.68253i

0.590972

+

1.02359i

0

0.500000

−

0.866025i

−0.942820

0

−0.282630

127.3

0.849814

−

1.47192i

0

−0.444368

−

0.769668i

−1.79418

−

3.10761i

0

0.500000

−

0.866025i

1.88874

0

−6.09888

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.2.f.a		6
3.b	odd	2	1		63.2.f.b	✓	6
4.b	odd	2	1		3024.2.r.g		6
7.b	odd	2	1		1323.2.f.c		6
7.c	even	3	1		1323.2.g.c		6
7.c	even	3	1		1323.2.h.d		6
7.d	odd	6	1		1323.2.g.b		6
7.d	odd	6	1		1323.2.h.e		6
9.c	even	3	1	inner	189.2.f.a		6
9.c	even	3	1		567.2.a.g		3
9.d	odd	6	1		63.2.f.b	✓	6
9.d	odd	6	1		567.2.a.d		3
12.b	even	2	1		1008.2.r.k		6
21.c	even	2	1		441.2.f.d		6
21.g	even	6	1		441.2.g.d		6
21.g	even	6	1		441.2.h.b		6
21.h	odd	6	1		441.2.g.e		6
21.h	odd	6	1		441.2.h.c		6
36.f	odd	6	1		3024.2.r.g		6
36.f	odd	6	1		9072.2.a.cd		3
36.h	even	6	1		1008.2.r.k		6
36.h	even	6	1		9072.2.a.bq		3
63.g	even	3	1		1323.2.h.d		6
63.h	even	3	1		1323.2.g.c		6
63.i	even	6	1		441.2.g.d		6
63.j	odd	6	1		441.2.g.e		6
63.k	odd	6	1		1323.2.h.e		6
63.l	odd	6	1		1323.2.f.c		6
63.l	odd	6	1		3969.2.a.p		3
63.n	odd	6	1		441.2.h.c		6
63.o	even	6	1		441.2.f.d		6
63.o	even	6	1		3969.2.a.m		3
63.s	even	6	1		441.2.h.b		6
63.t	odd	6	1		1323.2.g.b		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.f.b	✓	6	3.b	odd	2	1
63.2.f.b	✓	6	9.d	odd	6	1
189.2.f.a		6	1.a	even	1	1	trivial
189.2.f.a		6	9.c	even	3	1	inner
441.2.f.d		6	21.c	even	2	1
441.2.f.d		6	63.o	even	6	1
441.2.g.d		6	21.g	even	6	1
441.2.g.d		6	63.i	even	6	1
441.2.g.e		6	21.h	odd	6	1
441.2.g.e		6	63.j	odd	6	1
441.2.h.b		6	21.g	even	6	1
441.2.h.b		6	63.s	even	6	1
441.2.h.c		6	21.h	odd	6	1
441.2.h.c		6	63.n	odd	6	1
567.2.a.d		3	9.d	odd	6	1
567.2.a.g		3	9.c	even	3	1
1008.2.r.k		6	12.b	even	2	1
1008.2.r.k		6	36.h	even	6	1
1323.2.f.c		6	7.b	odd	2	1
1323.2.f.c		6	63.l	odd	6	1
1323.2.g.b		6	7.d	odd	6	1
1323.2.g.b		6	63.t	odd	6	1
1323.2.g.c		6	7.c	even	3	1
1323.2.g.c		6	63.h	even	3	1
1323.2.h.d		6	7.c	even	3	1
1323.2.h.d		6	63.g	even	3	1
1323.2.h.e		6	7.d	odd	6	1
1323.2.h.e		6	63.k	odd	6	1
3024.2.r.g		6	4.b	odd	2	1
3024.2.r.g		6	36.f	odd	6	1
3969.2.a.m		3	63.o	even	6	1
3969.2.a.p		3	63.l	odd	6	1
9072.2.a.bq		3	36.h	even	6	1
9072.2.a.cd		3	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 + T^5 + 5*T^4 - 2*T^3 + 17*T^2 + 4*T + 1
$3$	\( T^{6} \)
$5$	T^6 + 5*T^5 + 23*T^4 + 32*T^3 + 59*T^2 - 22*T + 121
$7$	\( (T^{2} - T + 1)^{3} \)
$11$	T^6 + 2*T^5 + 23*T^4 + 56*T^3 + 455*T^2 + 893*T + 2209