Newform orbit 405.4.e.c

• Label: 405.4.e.c
• Level: $405$
• Weight: $4$
• Character orbit: 405.e
• Analytic conductor: $23.896$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.8957735523\)
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 5)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

136.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.00000

+

3.46410i

0

−4.00000

−

6.92820i

−2.50000

−

4.33013i

0

−3.00000

+

5.19615i

0

0

20.0000

271.1

−2.00000

−

3.46410i

0

−4.00000

+

6.92820i

−2.50000

+

4.33013i

0

−3.00000

−

5.19615i

0

0

20.0000

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	405.4.e.c		2
3.b	odd	2	1		405.4.e.l		2
9.c	even	3	1		45.4.a.d		1
9.c	even	3	1	inner	405.4.e.c		2
9.d	odd	6	1		5.4.a.a	✓	1
9.d	odd	6	1		405.4.e.l		2
36.f	odd	6	1		720.4.a.u		1
36.h	even	6	1		80.4.a.d		1
45.h	odd	6	1		25.4.a.c		1
45.j	even	6	1		225.4.a.b		1
45.k	odd	12	2		225.4.b.c		2
45.l	even	12	2		25.4.b.a		2
63.i	even	6	1		245.4.e.g		2
63.j	odd	6	1		245.4.e.f		2
63.l	odd	6	1		2205.4.a.q		1
63.n	odd	6	1		245.4.e.f		2
63.o	even	6	1		245.4.a.a		1
63.s	even	6	1		245.4.e.g		2
72.j	odd	6	1		320.4.a.g		1
72.l	even	6	1		320.4.a.h		1
99.g	even	6	1		605.4.a.d		1
117.n	odd	6	1		845.4.a.b		1
144.u	even	12	2		1280.4.d.l		2
144.w	odd	12	2		1280.4.d.e		2
153.i	odd	6	1		1445.4.a.a		1
171.l	even	6	1		1805.4.a.h		1
180.n	even	6	1		400.4.a.m		1
180.v	odd	12	2		400.4.c.k		2
315.z	even	6	1		1225.4.a.k		1
360.bd	even	6	1		1600.4.a.s		1
360.bh	odd	6	1		1600.4.a.bi		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	9.d	odd	6	1
25.4.a.c		1	45.h	odd	6	1
25.4.b.a		2	45.l	even	12	2
45.4.a.d		1	9.c	even	3	1
80.4.a.d		1	36.h	even	6	1
225.4.a.b		1	45.j	even	6	1
225.4.b.c		2	45.k	odd	12	2
245.4.a.a		1	63.o	even	6	1
245.4.e.f		2	63.j	odd	6	1
245.4.e.f		2	63.n	odd	6	1
245.4.e.g		2	63.i	even	6	1
245.4.e.g		2	63.s	even	6	1
320.4.a.g		1	72.j	odd	6	1
320.4.a.h		1	72.l	even	6	1
400.4.a.m		1	180.n	even	6	1
400.4.c.k		2	180.v	odd	12	2
405.4.e.c		2	1.a	even	1	1	trivial
405.4.e.c		2	9.c	even	3	1	inner
405.4.e.l		2	3.b	odd	2	1
405.4.e.l		2	9.d	odd	6	1
605.4.a.d		1	99.g	even	6	1
720.4.a.u		1	36.f	odd	6	1
845.4.a.b		1	117.n	odd	6	1
1225.4.a.k		1	315.z	even	6	1
1280.4.d.e		2	144.w	odd	12	2
1280.4.d.l		2	144.u	even	12	2
1445.4.a.a		1	153.i	odd	6	1
1600.4.a.s		1	360.bd	even	6	1
1600.4.a.bi		1	360.bh	odd	6	1
1805.4.a.h		1	171.l	even	6	1
2205.4.a.q		1	63.l	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

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

\( T_{7}^{2} + 6T_{7} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 4T + 16 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 5T + 25 \)
$7$	\( T^{2} + 6T + 36 \)
$11$	\( T^{2} - 32T + 1024 \)