Newform orbit 405.2.e.f

• Label: 405.2.e.f
• Level: $405$
• Weight: $2$
• Character orbit: 405.e
• Analytic conductor: $3.234$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.23394128186\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
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.

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

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

0.500000

−

0.866025i

0

0.500000

+

0.866025i

−0.500000

−

0.866025i

0

0

3.00000

0

−1.00000

271.1

0.500000

+

0.866025i

0

0.500000

−

0.866025i

−0.500000

+

0.866025i

0

0

3.00000

0

−1.00000

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	405.2.e.f		2
3.b	odd	2	1		405.2.e.c		2
9.c	even	3	1		15.2.a.a	✓	1
9.c	even	3	1	inner	405.2.e.f		2
9.d	odd	6	1		45.2.a.a		1
9.d	odd	6	1		405.2.e.c		2
36.f	odd	6	1		240.2.a.d		1
36.h	even	6	1		720.2.a.c		1
45.h	odd	6	1		225.2.a.b		1
45.j	even	6	1		75.2.a.b		1
45.k	odd	12	2		75.2.b.b		2
45.l	even	12	2		225.2.b.b		2
63.g	even	3	1		735.2.i.e		2
63.h	even	3	1		735.2.i.e		2
63.k	odd	6	1		735.2.i.d		2
63.l	odd	6	1		735.2.a.c		1
63.o	even	6	1		2205.2.a.i		1
63.t	odd	6	1		735.2.i.d		2
72.j	odd	6	1		2880.2.a.y		1
72.l	even	6	1		2880.2.a.bc		1
72.n	even	6	1		960.2.a.l		1
72.p	odd	6	1		960.2.a.a		1
99.g	even	6	1		5445.2.a.c		1
99.h	odd	6	1		1815.2.a.d		1
117.n	odd	6	1		7605.2.a.g		1
117.t	even	6	1		2535.2.a.j		1
144.v	odd	12	2		3840.2.k.r		2
144.x	even	12	2		3840.2.k.m		2
153.h	even	6	1		4335.2.a.c		1
171.o	odd	6	1		5415.2.a.j		1
180.n	even	6	1		3600.2.a.u		1
180.p	odd	6	1		1200.2.a.e		1
180.v	odd	12	2		3600.2.f.e		2
180.x	even	12	2		1200.2.f.h		2
207.f	odd	6	1		7935.2.a.d		1
315.bg	odd	6	1		3675.2.a.j		1
360.z	odd	6	1		4800.2.a.bz		1
360.bk	even	6	1		4800.2.a.t		1
360.bo	even	12	2		4800.2.f.c		2
360.bu	odd	12	2		4800.2.f.bf		2
495.o	odd	6	1		9075.2.a.g		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.2.a.a	✓	1	9.c	even	3	1
45.2.a.a		1	9.d	odd	6	1
75.2.a.b		1	45.j	even	6	1
75.2.b.b		2	45.k	odd	12	2
225.2.a.b		1	45.h	odd	6	1
225.2.b.b		2	45.l	even	12	2
240.2.a.d		1	36.f	odd	6	1
405.2.e.c		2	3.b	odd	2	1
405.2.e.c		2	9.d	odd	6	1
405.2.e.f		2	1.a	even	1	1	trivial
405.2.e.f		2	9.c	even	3	1	inner
720.2.a.c		1	36.h	even	6	1
735.2.a.c		1	63.l	odd	6	1
735.2.i.d		2	63.k	odd	6	1
735.2.i.d		2	63.t	odd	6	1
735.2.i.e		2	63.g	even	3	1
735.2.i.e		2	63.h	even	3	1
960.2.a.a		1	72.p	odd	6	1
960.2.a.l		1	72.n	even	6	1
1200.2.a.e		1	180.p	odd	6	1
1200.2.f.h		2	180.x	even	12	2
1815.2.a.d		1	99.h	odd	6	1
2205.2.a.i		1	63.o	even	6	1
2535.2.a.j		1	117.t	even	6	1
2880.2.a.y		1	72.j	odd	6	1
2880.2.a.bc		1	72.l	even	6	1
3600.2.a.u		1	180.n	even	6	1
3600.2.f.e		2	180.v	odd	12	2
3675.2.a.j		1	315.bg	odd	6	1
3840.2.k.m		2	144.x	even	12	2
3840.2.k.r		2	144.v	odd	12	2
4335.2.a.c		1	153.h	even	6	1
4800.2.a.t		1	360.bk	even	6	1
4800.2.a.bz		1	360.z	odd	6	1
4800.2.f.c		2	360.bo	even	12	2
4800.2.f.bf		2	360.bu	odd	12	2
5415.2.a.j		1	171.o	odd	6	1
5445.2.a.c		1	99.g	even	6	1
7605.2.a.g		1	117.n	odd	6	1
7935.2.a.d		1	207.f	odd	6	1
9075.2.a.g		1	495.o	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \)

\( T_{7} \)

\( T_{11}^{2} - 4T_{11} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T + 16 \)