Newform orbit 810.4.e.c

• Label: 810.4.e.c
• Level: $810$
• Weight: $4$
• Character orbit: 810.e
• Analytic conductor: $47.792$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(47.7915471046\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

271.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

0

−2.00000

+

3.46410i

−2.50000

+

4.33013i

0

2.00000

+

3.46410i

8.00000

0

10.0000

541.1

−1.00000

+

1.73205i

0

−2.00000

−

3.46410i

−2.50000

−

4.33013i

0

2.00000

−

3.46410i

8.00000

0

10.0000

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	810.4.e.c		2
3.b	odd	2	1		810.4.e.w		2
9.c	even	3	1		10.4.a.a	✓	1
9.c	even	3	1	inner	810.4.e.c		2
9.d	odd	6	1		90.4.a.a		1
9.d	odd	6	1		810.4.e.w		2
36.f	odd	6	1		80.4.a.f		1
36.h	even	6	1		720.4.a.j		1
45.h	odd	6	1		450.4.a.q		1
45.j	even	6	1		50.4.a.c		1
45.k	odd	12	2		50.4.b.a		2
45.l	even	12	2		450.4.c.d		2
63.g	even	3	1		490.4.e.i		2
63.h	even	3	1		490.4.e.i		2
63.k	odd	6	1		490.4.e.a		2
63.l	odd	6	1		490.4.a.o		1
63.t	odd	6	1		490.4.e.a		2
72.n	even	6	1		320.4.a.m		1
72.p	odd	6	1		320.4.a.b		1
99.h	odd	6	1		1210.4.a.b		1
117.t	even	6	1		1690.4.a.a		1
144.v	odd	12	2		1280.4.d.g		2
144.x	even	12	2		1280.4.d.j		2
180.p	odd	6	1		400.4.a.b		1
180.x	even	12	2		400.4.c.c		2
315.bg	odd	6	1		2450.4.a.b		1
360.z	odd	6	1		1600.4.a.bx		1
360.bk	even	6	1		1600.4.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.4.a.a	✓	1	9.c	even	3	1
50.4.a.c		1	45.j	even	6	1
50.4.b.a		2	45.k	odd	12	2
80.4.a.f		1	36.f	odd	6	1
90.4.a.a		1	9.d	odd	6	1
320.4.a.b		1	72.p	odd	6	1
320.4.a.m		1	72.n	even	6	1
400.4.a.b		1	180.p	odd	6	1
400.4.c.c		2	180.x	even	12	2
450.4.a.q		1	45.h	odd	6	1
450.4.c.d		2	45.l	even	12	2
490.4.a.o		1	63.l	odd	6	1
490.4.e.a		2	63.k	odd	6	1
490.4.e.a		2	63.t	odd	6	1
490.4.e.i		2	63.g	even	3	1
490.4.e.i		2	63.h	even	3	1
720.4.a.j		1	36.h	even	6	1
810.4.e.c		2	1.a	even	1	1	trivial
810.4.e.c		2	9.c	even	3	1	inner
810.4.e.w		2	3.b	odd	2	1
810.4.e.w		2	9.d	odd	6	1
1210.4.a.b		1	99.h	odd	6	1
1280.4.d.g		2	144.v	odd	12	2
1280.4.d.j		2	144.x	even	12	2
1600.4.a.d		1	360.bk	even	6	1
1600.4.a.bx		1	360.z	odd	6	1
1690.4.a.a		1	117.t	even	6	1
2450.4.a.b		1	315.bg	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 16 \)

\( T_{11}^{2} + 12T_{11} + 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 5T + 25 \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} + 12T + 144 \)