LMFDB - Newform orbit 4704.2.a.t

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4704,2,Mod(1,4704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4704.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4704 = 2^{5} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4704.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,-2,0,0,0,1,0,-4,0,2,0,-2,0,6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

f(q)

=

q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} - q^{25} + q^{27} + 2 q^{29} + 4 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100})

q + q^3 - 2 * q^5 + q^9 - 4 * q^11 + 2 * q^13 - 2 * q^15 + 6 * q^17 - 4 * q^19 - q^25 + q^27 + 2 * q^29 + 4 * q^31 - 4 * q^33 - 2 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^45 + 8 * q^47 + 6 * q^51 + 10 * q^53 + 8 * q^55 - 4 * q^57 - 4 * q^59 - 6 * q^61 - 4 * q^65 - 4 * q^67 + 16 * q^71 + 6 * q^73 - q^75 - 4 * q^79 + q^81 + 12 * q^83 - 12 * q^85 + 2 * q^87 - 10 * q^89 + 4 * q^93 + 8 * q^95 + 14 * q^97 - 4 * q^99

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

1.00000

−2.00000

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4704.2.a.t		1
4.b	odd	2	1		4704.2.a.e		1
7.b	odd	2	1		96.2.a.a	✓	1
8.b	even	2	1		9408.2.a.bj		1
8.d	odd	2	1		9408.2.a.ct		1
21.c	even	2	1		288.2.a.c		1
28.d	even	2	1		96.2.a.b	yes	1
35.c	odd	2	1		2400.2.a.r		1
35.f	even	4	2		2400.2.f.a		2
56.e	even	2	1		192.2.a.a		1
56.h	odd	2	1		192.2.a.c		1
63.l	odd	6	2		2592.2.i.b		2
63.o	even	6	2		2592.2.i.q		2
84.h	odd	2	1		288.2.a.b		1
105.g	even	2	1		7200.2.a.e		1
105.k	odd	4	2		7200.2.f.x		2
112.j	even	4	2		768.2.d.h		2
112.l	odd	4	2		768.2.d.a		2
140.c	even	2	1		2400.2.a.q		1
140.j	odd	4	2		2400.2.f.r		2
168.e	odd	2	1		576.2.a.g		1
168.i	even	2	1		576.2.a.h		1
252.s	odd	6	2		2592.2.i.w		2
252.bi	even	6	2		2592.2.i.h		2
280.c	odd	2	1		4800.2.a.f		1
280.n	even	2	1		4800.2.a.co		1
280.s	even	4	2		4800.2.f.bh		2
280.y	odd	4	2		4800.2.f.e		2
336.v	odd	4	2		2304.2.d.s		2
336.y	even	4	2		2304.2.d.c		2
420.o	odd	2	1		7200.2.a.bx		1
420.w	even	4	2		7200.2.f.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.a.a	✓	1	7.b	odd	2	1
96.2.a.b	yes	1	28.d	even	2	1
192.2.a.a		1	56.e	even	2	1
192.2.a.c		1	56.h	odd	2	1
288.2.a.b		1	84.h	odd	2	1
288.2.a.c		1	21.c	even	2	1
576.2.a.g		1	168.e	odd	2	1
576.2.a.h		1	168.i	even	2	1
768.2.d.a		2	112.l	odd	4	2
768.2.d.h		2	112.j	even	4	2
2304.2.d.c		2	336.y	even	4	2
2304.2.d.s		2	336.v	odd	4	2
2400.2.a.q		1	140.c	even	2	1
2400.2.a.r		1	35.c	odd	2	1
2400.2.f.a		2	35.f	even	4	2
2400.2.f.r		2	140.j	odd	4	2
2592.2.i.b		2	63.l	odd	6	2
2592.2.i.h		2	252.bi	even	6	2
2592.2.i.q		2	63.o	even	6	2
2592.2.i.w		2	252.s	odd	6	2
4704.2.a.e		1	4.b	odd	2	1
4704.2.a.t		1	1.a	even	1	1	trivial
4800.2.a.f		1	280.c	odd	2	1
4800.2.a.co		1	280.n	even	2	1
4800.2.f.e		2	280.y	odd	4	2
4800.2.f.bh		2	280.s	even	4	2
7200.2.a.e		1	105.g	even	2	1
7200.2.a.bx		1	420.o	odd	2	1
7200.2.f.f		2	420.w	even	4	2
7200.2.f.x		2	105.k	odd	4	2
9408.2.a.bj		1	8.b	even	2	1
9408.2.a.ct		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4704))$ :

T_{5} + 2

T5 + 2

T_{11} + 4

T11 + 4

T_{13} - 2

T13 - 2

T_{19} + 4

T19 + 4

T_{31} - 4

T31 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 1$ T - 1
$5$	$T + 2$ T + 2
$7$	$T$ T
$11$	$T + 4$ T + 4
$13$	$T - 2$ T - 2
$17$	$T - 6$ T - 6
$19$	$T + 4$ T + 4
$23$	$T$ T
$29$	$T - 2$ T - 2
$31$	$T - 4$ T - 4
$37$	$T + 2$ T + 2
$41$	$T + 2$ T + 2
$43$	$T + 4$ T + 4
$47$	$T - 8$ T - 8
$53$	$T - 10$ T - 10
$59$	$T + 4$ T + 4
$61$	$T + 6$ T + 6
$67$	$T + 4$ T + 4
$71$	$T - 16$ T - 16
$73$	$T - 6$ T - 6
$79$	$T + 4$ T + 4
$83$	$T - 12$ T - 12
$89$	$T + 10$ T + 10
$97$	$T - 14$ T - 14
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