LMFDB - Newform orbit 600.8.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,8,Mod(1,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-27,0,0,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$187.431015290$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
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 - 27 q^{3} - 120 q^{7} + 729 q^{9} - 7196 q^{11} + 9626 q^{13} - 18674 q^{17} + 7004 q^{19} + 3240 q^{21} + 63704 q^{23} - 19683 q^{27} + 29334 q^{29} + 87968 q^{31} + 194292 q^{33} - 227982 q^{37} - 259902 q^{39}+ \cdots - 5245884 q^{99}+O(q^{100})

q - 27 * q^3 - 120 * q^7 + 729 * q^9 - 7196 * q^11 + 9626 * q^13 - 18674 * q^17 + 7004 * q^19 + 3240 * q^21 + 63704 * q^23 - 19683 * q^27 + 29334 * q^29 + 87968 * q^31 + 194292 * q^33 - 227982 * q^37 - 259902 * q^39 - 160806 * q^41 - 136132 * q^43 + 1206960 * q^47 - 809143 * q^49 + 504198 * q^51 + 398786 * q^53 - 189108 * q^57 + 1152436 * q^59 - 2070602 * q^61 - 87480 * q^63 + 4073428 * q^67 - 1720008 * q^69 - 383752 * q^71 - 3006010 * q^73 + 863520 * q^77 - 4948112 * q^79 + 531441 * q^81 + 9163492 * q^83 - 792018 * q^87 + 7304106 * q^89 - 1155120 * q^91 - 2375136 * q^93 + 690526 * q^97 - 5245884 * 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

−27.0000

−120.000

729.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$5$	$+1$

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	600.8.a.b		1
5.b	even	2	1		24.8.a.b	✓	1
5.c	odd	4	2		600.8.f.a		2
15.d	odd	2	1		72.8.a.e		1
20.d	odd	2	1		48.8.a.a		1
40.e	odd	2	1		192.8.a.p		1
40.f	even	2	1		192.8.a.h		1
60.h	even	2	1		144.8.a.k		1
120.i	odd	2	1		576.8.a.c		1
120.m	even	2	1		576.8.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.8.a.b	✓	1	5.b	even	2	1
48.8.a.a		1	20.d	odd	2	1
72.8.a.e		1	15.d	odd	2	1
144.8.a.k		1	60.h	even	2	1
192.8.a.h		1	40.f	even	2	1
192.8.a.p		1	40.e	odd	2	1
576.8.a.b		1	120.m	even	2	1
576.8.a.c		1	120.i	odd	2	1
600.8.a.b		1	1.a	even	1	1	trivial
600.8.f.a		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7} + 120$ T7 + 120 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(600))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 27$ T + 27
$5$	$T$ T
$7$	$T + 120$ T + 120
$11$	$T + 7196$ T + 7196
$13$	$T - 9626$ T - 9626
$17$	$T + 18674$ T + 18674
$19$	$T - 7004$ T - 7004
$23$	$T - 63704$ T - 63704
$29$	$T - 29334$ T - 29334
$31$	$T - 87968$ T - 87968
$37$	$T + 227982$ T + 227982
$41$	$T + 160806$ T + 160806
$43$	$T + 136132$ T + 136132
$47$	$T - 1206960$ T - 1206960
$53$	$T - 398786$ T - 398786
$59$	$T - 1152436$ T - 1152436
$61$	$T + 2070602$ T + 2070602
$67$	$T - 4073428$ T - 4073428
$71$	$T + 383752$ T + 383752
$73$	$T + 3006010$ T + 3006010
$79$	$T + 4948112$ T + 4948112
$83$	$T - 9163492$ T - 9163492
$89$	$T - 7304106$ T - 7304106
$97$	$T - 690526$ T - 690526
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