Newform orbit 810.3.j.c

• Label: 810.3.j.c
• Level: $810$
• Weight: $3$
• Character orbit: 810.j
• Analytic conductor: $22.071$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.0709014132\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.443364212736.6
Defining polynomial:	\( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) :

\(\beta_{1}\)	\(=\)	\( ( -16\nu^{6} + 175\nu^{4} - 2800\nu^{2} + 20736 ) / 14175 \)
\(\beta_{2}\)	\(=\)	\( ( 31\nu^{7} - 1225\nu^{5} + 5425\nu^{3} - 40176\nu ) / 127575 \)
\(\beta_{3}\)	\(=\)	\( ( -47\nu^{6} + 1400\nu^{4} - 8225\nu^{2} + 60912 ) / 14175 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 3079\nu ) / 1575 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{6} - 104 ) / 175 \)
\(\beta_{6}\)	\(=\)	\( ( -16\nu^{7} + 175\nu^{5} - 775\nu^{3} + 6561\nu ) / 18225 \)
\(\beta_{7}\)	\(=\)	\( ( -319\nu^{7} + 4375\nu^{5} - 55825\nu^{3} + 413424\nu ) / 127575 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

269.1

−2.17132	−	2.07011i
2.87843	+	0.845366i
2.17132	+	2.07011i
−2.87843	−	0.845366i
−2.17132	+	2.07011i
2.87843	−	0.845366i
2.17132	−	2.07011i
−2.87843	+	0.845366i

−0.707107

+

1.22474i

0

−1.00000

−

1.73205i

−2.15650

+

4.51104i

0

5.04975

+

2.91548i

2.82843

0

−4.00000

−

5.83095i

269.2

−0.707107

+

1.22474i

0

−1.00000

−

1.73205i

4.98493

+

0.387937i

0

−5.04975

−

2.91548i

2.82843

0

−4.00000

+

5.83095i

269.3

0.707107

−

1.22474i

0

−1.00000

−

1.73205i

−4.98493

−

0.387937i

0

−5.04975

−

2.91548i

−2.82843

0

−4.00000

+

5.83095i

269.4

0.707107

−

1.22474i

0

−1.00000

−

1.73205i

2.15650

−

4.51104i

0

5.04975

+

2.91548i

−2.82843

0

−4.00000

−

5.83095i

539.1

−0.707107

−

1.22474i

0

−1.00000

+

1.73205i

−2.15650

−

4.51104i

0

5.04975

−

2.91548i

2.82843

0

−4.00000

+

5.83095i

539.2

−0.707107

−

1.22474i

0

−1.00000

+

1.73205i

4.98493

−

0.387937i

0

−5.04975

+

2.91548i

2.82843

0

−4.00000

−

5.83095i

539.3

0.707107

+

1.22474i

0

−1.00000

+

1.73205i

−4.98493

+

0.387937i

0

−5.04975

+

2.91548i

−2.82843

0

−4.00000

−

5.83095i

539.4

0.707107

+

1.22474i

0

−1.00000

+

1.73205i

2.15650

+

4.51104i

0

5.04975

−

2.91548i

−2.82843

0

−4.00000

+

5.83095i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
15.d	odd	2	1	inner
45.h	odd	6	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.3.j.c		8
3.b	odd	2	1	inner	810.3.j.c		8
5.b	even	2	1	inner	810.3.j.c		8
9.c	even	3	1		30.3.b.a	✓	4
9.c	even	3	1	inner	810.3.j.c		8
9.d	odd	6	1		30.3.b.a	✓	4
9.d	odd	6	1	inner	810.3.j.c		8
15.d	odd	2	1	inner	810.3.j.c		8
36.f	odd	6	1		240.3.c.c		4
36.h	even	6	1		240.3.c.c		4
45.h	odd	6	1		30.3.b.a	✓	4
45.h	odd	6	1	inner	810.3.j.c		8
45.j	even	6	1		30.3.b.a	✓	4
45.j	even	6	1	inner	810.3.j.c		8
45.k	odd	12	2		150.3.d.d		4
45.l	even	12	2		150.3.d.d		4
72.j	odd	6	1		960.3.c.f		4
72.l	even	6	1		960.3.c.e		4
72.n	even	6	1		960.3.c.f		4
72.p	odd	6	1		960.3.c.e		4
180.n	even	6	1		240.3.c.c		4
180.p	odd	6	1		240.3.c.c		4
180.v	odd	12	2		1200.3.l.t		4
180.x	even	12	2		1200.3.l.t		4
360.z	odd	6	1		960.3.c.e		4
360.bd	even	6	1		960.3.c.e		4
360.bh	odd	6	1		960.3.c.f		4
360.bk	even	6	1		960.3.c.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.3.b.a	✓	4	9.c	even	3	1
30.3.b.a	✓	4	9.d	odd	6	1
30.3.b.a	✓	4	45.h	odd	6	1
30.3.b.a	✓	4	45.j	even	6	1
150.3.d.d		4	45.k	odd	12	2
150.3.d.d		4	45.l	even	12	2
240.3.c.c		4	36.f	odd	6	1
240.3.c.c		4	36.h	even	6	1
240.3.c.c		4	180.n	even	6	1
240.3.c.c		4	180.p	odd	6	1
810.3.j.c		8	1.a	even	1	1	trivial
810.3.j.c		8	3.b	odd	2	1	inner
810.3.j.c		8	5.b	even	2	1	inner
810.3.j.c		8	9.c	even	3	1	inner
810.3.j.c		8	9.d	odd	6	1	inner
810.3.j.c		8	15.d	odd	2	1	inner
810.3.j.c		8	45.h	odd	6	1	inner
810.3.j.c		8	45.j	even	6	1	inner
960.3.c.e		4	72.l	even	6	1
960.3.c.e		4	72.p	odd	6	1
960.3.c.e		4	360.z	odd	6	1
960.3.c.e		4	360.bd	even	6	1
960.3.c.f		4	72.j	odd	6	1
960.3.c.f		4	72.n	even	6	1
960.3.c.f		4	360.bh	odd	6	1
960.3.c.f		4	360.bk	even	6	1
1200.3.l.t		4	180.v	odd	12	2
1200.3.l.t		4	180.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} - 34T_{7}^{2} + 1156 \)

\( T_{17}^{2} - 128 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + 2 T^{2} + 4)^{2} \)
$3$	\( T^{8} \)
$5$	T^8 - 18*T^6 - 301*T^4 - 11250*T^2 + 390625
$7$	\( (T^{4} - 34 T^{2} + 1156)^{2} \)
$11$	\( (T^{4} - 272 T^{2} + 73984)^{2} \)