Newform orbit 810.4.e.k

• Label: 810.4.e.k
• 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 270)
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.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 22 \zeta_{6} + 22) q^{7} + 8 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 22 q^{7} + 16 q^{8}+O(q^{10}) \)

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

11.0000

+

19.0526i

8.00000

0

−10.0000

541.1

−1.00000

+

1.73205i

0

−2.00000

−

3.46410i

2.50000

+

4.33013i

0

11.0000

−

19.0526i

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.k		2
3.b	odd	2	1		810.4.e.s		2
9.c	even	3	1		270.4.a.g	yes	1
9.c	even	3	1	inner	810.4.e.k		2
9.d	odd	6	1		270.4.a.c	✓	1
9.d	odd	6	1		810.4.e.s		2
36.f	odd	6	1		2160.4.a.i		1
36.h	even	6	1		2160.4.a.r		1
45.h	odd	6	1		1350.4.a.z		1
45.j	even	6	1		1350.4.a.l		1
45.k	odd	12	2		1350.4.c.i		2
45.l	even	12	2		1350.4.c.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.c	✓	1	9.d	odd	6	1
270.4.a.g	yes	1	9.c	even	3	1
810.4.e.k		2	1.a	even	1	1	trivial
810.4.e.k		2	9.c	even	3	1	inner
810.4.e.s		2	3.b	odd	2	1
810.4.e.s		2	9.d	odd	6	1
1350.4.a.l		1	45.j	even	6	1
1350.4.a.z		1	45.h	odd	6	1
1350.4.c.i		2	45.k	odd	12	2
1350.4.c.l		2	45.l	even	12	2
2160.4.a.i		1	36.f	odd	6	1
2160.4.a.r		1	36.h	even	6	1

Hecke kernels

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

\( T_{7}^{2} - 22T_{7} + 484 \)

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 5T + 25 \)
$7$	\( T^{2} - 22T + 484 \)
$11$	\( T^{2} - 12T + 144 \)