LMFDB - Newform orbit 48.22.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$134.149125258$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
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 + 59049 q^{3} - 41512770 q^{5} - 538429808 q^{7} + 3486784401 q^{9}+O(q^{10}) $

$ q + 59049 q^{3} - 41512770 q^{5} - 538429808 q^{7} + 3486784401 q^{9} + 64113040188 q^{11} - 130980107986 q^{13} - 2451287555730 q^{15} + 8242029723618 q^{17} - 13492101753020 q^{19} - 31793741732592 q^{21} + 233184825844776 q^{23} + 12\!\cdots\!75 q^{25}+ \cdots + 22\!\cdots\!88 q^{99}+O(q^{100}) $

q + 59049 * q^3 - 41512770 * q^5 - 538429808 * q^7 + 3486784401 * q^9 + 64113040188 * q^11 - 130980107986 * q^13 - 2451287555730 * q^15 + 8242029723618 * q^17 - 13492101753020 * q^19 - 31793741732592 * q^21 + 233184825844776 * q^23 + 1246472914869775 * q^25 + 205891132094649 * q^27 - 2024562031123770 * q^29 + 6869194988701768 * q^31 + 3785810910061212 * q^33 + 22351712780648160 * q^35 + 3443998107027638 * q^37 - 7734244396465314 * q^39 - 21842403084625158 * q^41 + 71792816814133756 * q^43 - 144746078878300770 * q^45 - 283544719418655648 * q^47 - 268639205940367143 * q^49 + 486683613149919282 * q^51 - 2172285419049898146 * q^53 - 2661509891325200760 * q^55 - 796695116414077980 * q^57 - 1534831476719068260 * q^59 + 4311589520797626062 * q^61 - 1877388655567825008 * q^63 + 5437347097397981220 * q^65 - 9243910904037307868 * q^67 + 13769330781308178024 * q^69 + 20387361256404760728 * q^71 + 16617754439328636074 * q^73 + 73602979150145343975 * q^75 - 34520371918721123904 * q^77 - 67940304745507627880 * q^79 + 12157665459056928801 * q^81 - 39503732340682314684 * q^83 - 342149484249717601860 * q^85 - 119548363375827494730 * q^87 + 41611676186839694490 * q^89 + 70523594394721246688 * q^91 + 405619094887850698632 * q^93 + 560094516889716065400 * q^95 + 57181473208903260098 * q^97 + 223548348428204507388 * q^99

Display 100 coefficients 10 coefficients

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

59049.0

−4.15128e7

−5.38430e8

3.48678e9

Atkin-Lehner signs

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

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	48.22.a.d		1
4.b	odd	2	1		3.22.a.b	✓	1
12.b	even	2	1		9.22.a.a		1
20.d	odd	2	1		75.22.a.a		1
20.e	even	4	2		75.22.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.22.a.b	✓	1	4.b	odd	2	1
9.22.a.a		1	12.b	even	2	1
48.22.a.d		1	1.a	even	1	1	trivial
75.22.a.a		1	20.d	odd	2	1
75.22.b.b		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 41512770 $ T5 + 41512770 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(48))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 59049 $ T - 59049
$5$	$ T + 41512770 $ T + 41512770
$7$	$ T + 538429808 $ T + 538429808
$11$	$ T - 64113040188 $ T - 64113040188
$13$	$ T + 130980107986 $ T + 130980107986
$17$	$ T - 8242029723618 $ T - 8242029723618
$19$	$ T + 13492101753020 $ T + 13492101753020
$23$	$ T - 233184825844776 $ T - 233184825844776
$29$	$ T + 2024562031123770 $ T + 2024562031123770
$31$	$ T - 6869194988701768 $ T - 6869194988701768
$37$	$ T - 3443998107027638 $ T - 3443998107027638
$41$	$ T + 21\!\cdots\!58 $ T + 21842403084625158
$43$	$ T - 71\!\cdots\!56 $ T - 71792816814133756
$47$	$ T + 28\!\cdots\!48 $ T + 283544719418655648
$53$	$ T + 21\!\cdots\!46 $ T + 2172285419049898146
$59$	$ T + 15\!\cdots\!60 $ T + 1534831476719068260
$61$	$ T - 43\!\cdots\!62 $ T - 4311589520797626062
$67$	$ T + 92\!\cdots\!68 $ T + 9243910904037307868
$71$	$ T - 20\!\cdots\!28 $ T - 20387361256404760728
$73$	$ T - 16\!\cdots\!74 $ T - 16617754439328636074
$79$	$ T + 67\!\cdots\!80 $ T + 67940304745507627880
$83$	$ T + 39\!\cdots\!84 $ T + 39503732340682314684
$89$	$ T - 41\!\cdots\!90 $ T - 41611676186839694490
$97$	$ T - 57\!\cdots\!98 $ T - 57181473208903260098
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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