LMFDB - Newform orbit 1600.4.a.m

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,4,Mod(1,1600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1600.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-5,0,0,0,2,0,-2,0,-39,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$94.4030560092$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 200)
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 - 5 q^{3} + 2 q^{7} - 2 q^{9} - 39 q^{11} - 84 q^{13} - 61 q^{17} - 151 q^{19} - 10 q^{21} - 58 q^{23} + 145 q^{27} - 192 q^{29} - 18 q^{31} + 195 q^{33} + 138 q^{37} + 420 q^{39} + 229 q^{41} + 164 q^{43}+ \cdots + 78 q^{99}+O(q^{100})

q - 5 * q^3 + 2 * q^7 - 2 * q^9 - 39 * q^11 - 84 * q^13 - 61 * q^17 - 151 * q^19 - 10 * q^21 - 58 * q^23 + 145 * q^27 - 192 * q^29 - 18 * q^31 + 195 * q^33 + 138 * q^37 + 420 * q^39 + 229 * q^41 + 164 * q^43 - 212 * q^47 - 339 * q^49 + 305 * q^51 - 578 * q^53 + 755 * q^57 + 336 * q^59 - 858 * q^61 - 4 * q^63 + 209 * q^67 + 290 * q^69 - 780 * q^71 - 403 * q^73 - 78 * q^77 - 230 * q^79 - 671 * q^81 + 1293 * q^83 + 960 * q^87 - 1369 * q^89 - 168 * q^91 + 90 * q^93 + 382 * q^97 + 78 * 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

−5.00000

2.00000

−2.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$

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	1600.4.a.m		1
4.b	odd	2	1		1600.4.a.bo		1
5.b	even	2	1		1600.4.a.bn		1
8.b	even	2	1		200.4.a.h	yes	1
8.d	odd	2	1		400.4.a.f		1
20.d	odd	2	1		1600.4.a.n		1
24.h	odd	2	1		1800.4.a.t		1
40.e	odd	2	1		400.4.a.q		1
40.f	even	2	1		200.4.a.c	✓	1
40.i	odd	4	2		200.4.c.d		2
40.k	even	4	2		400.4.c.g		2
120.i	odd	2	1		1800.4.a.p		1
120.w	even	4	2		1800.4.f.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.4.a.c	✓	1	40.f	even	2	1
200.4.a.h	yes	1	8.b	even	2	1
200.4.c.d		2	40.i	odd	4	2
400.4.a.f		1	8.d	odd	2	1
400.4.a.q		1	40.e	odd	2	1
400.4.c.g		2	40.k	even	4	2
1600.4.a.m		1	1.a	even	1	1	trivial
1600.4.a.n		1	20.d	odd	2	1
1600.4.a.bn		1	5.b	even	2	1
1600.4.a.bo		1	4.b	odd	2	1
1800.4.a.p		1	120.i	odd	2	1
1800.4.a.t		1	24.h	odd	2	1
1800.4.f.c		2	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1600))$ :

T_{3} + 5

T3 + 5

T_{7} - 2

T7 - 2

T_{11} + 39

T11 + 39

T_{13} + 84

T13 + 84

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 5$ T + 5
$5$	$T$ T
$7$	$T - 2$ T - 2
$11$	$T + 39$ T + 39
$13$	$T + 84$ T + 84
$17$	$T + 61$ T + 61
$19$	$T + 151$ T + 151
$23$	$T + 58$ T + 58
$29$	$T + 192$ T + 192
$31$	$T + 18$ T + 18
$37$	$T - 138$ T - 138
$41$	$T - 229$ T - 229
$43$	$T - 164$ T - 164
$47$	$T + 212$ T + 212
$53$	$T + 578$ T + 578
$59$	$T - 336$ T - 336
$61$	$T + 858$ T + 858
$67$	$T - 209$ T - 209
$71$	$T + 780$ T + 780
$73$	$T + 403$ T + 403
$79$	$T + 230$ T + 230
$83$	$T - 1293$ T - 1293
$89$	$T + 1369$ T + 1369
$97$	$T - 382$ T - 382
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