LMFDB - Newform orbit 18.16.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [18,16,Mod(1,18)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	18.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,128,0,16384,314490] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.6848309180$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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 + 128 q^{2} + 16384 q^{4} + 314490 q^{5} + 2025056 q^{7} + 2097152 q^{8} + 40254720 q^{10} - 110255052 q^{11} + 56047862 q^{13} + 259207168 q^{14} + 268435456 q^{16} + 1930104414 q^{17} + 2163188180 q^{19}+ \cdots - 82778842471296 q^{98}+O(q^{100}) $

q + 128 * q^2 + 16384 * q^4 + 314490 * q^5 + 2025056 * q^7 + 2097152 * q^8 + 40254720 * q^10 - 110255052 * q^11 + 56047862 * q^13 + 259207168 * q^14 + 268435456 * q^16 + 1930104414 * q^17 + 2163188180 * q^19 + 5152604160 * q^20 - 14112646656 * q^22 - 6228974472 * q^23 + 68386381975 * q^25 + 7174126336 * q^26 + 33178517504 * q^28 - 64743719070 * q^29 - 20237611048 * q^31 + 34359738368 * q^32 + 247053364992 * q^34 + 636859861440 * q^35 + 488967594446 * q^37 + 276888087040 * q^38 + 659533332480 * q^40 + 772359114198 * q^41 + 1306766329292 * q^43 - 1806418771968 * q^44 - 797308732416 * q^46 - 3351821491776 * q^47 - 646709706807 * q^49 + 8753456892800 * q^50 + 918288171008 * q^52 - 9387813393702 * q^53 - 34674111303480 * q^55 + 4246850240512 * q^56 - 8287196040960 * q^58 - 28930359275340 * q^59 + 42393077399702 * q^61 - 2590414214144 * q^62 + 4398046511104 * q^64 + 17626492120380 * q^65 - 52247243064364 * q^67 + 31622830718976 * q^68 + 81518062264320 * q^70 + 27194529024648 * q^71 - 91604195687878 * q^73 + 62587852089088 * q^74 + 35441675141120 * q^76 - 223272654582912 * q^77 + 62882111078120 * q^79 + 84420266557440 * q^80 + 98861966617344 * q^82 + 223567315949868 * q^83 + 606998537158860 * q^85 + 167266090149376 * q^86 - 231221602811904 * q^88 - 554198786115210 * q^89 + 113500059230272 * q^91 - 102055517749248 * q^92 - 429033150947328 * q^94 + 680301050728200 * q^95 - 1388870476877374 * q^97 - 82778842471296 * q^98

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

128.000

16384.0

314490.

2.02506e6

2.09715e6

4.02547e7

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	18.16.a.f		1
3.b	odd	2	1		6.16.a.a	✓	1
4.b	odd	2	1		144.16.a.o		1
12.b	even	2	1		48.16.a.c		1
15.d	odd	2	1		150.16.a.h		1
15.e	even	4	2		150.16.c.i		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.16.a.a	✓	1	3.b	odd	2	1
18.16.a.f		1	1.a	even	1	1	trivial
48.16.a.c		1	12.b	even	2	1
144.16.a.o		1	4.b	odd	2	1
150.16.a.h		1	15.d	odd	2	1
150.16.c.i		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 314490 $ T5 - 314490 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(18))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 128 $ T - 128
$3$	$ T $ T
$5$	$ T - 314490 $ T - 314490
$7$	$ T - 2025056 $ T - 2025056
$11$	$ T + 110255052 $ T + 110255052
$13$	$ T - 56047862 $ T - 56047862
$17$	$ T - 1930104414 $ T - 1930104414
$19$	$ T - 2163188180 $ T - 2163188180
$23$	$ T + 6228974472 $ T + 6228974472
$29$	$ T + 64743719070 $ T + 64743719070
$31$	$ T + 20237611048 $ T + 20237611048
$37$	$ T - 488967594446 $ T - 488967594446
$41$	$ T - 772359114198 $ T - 772359114198
$43$	$ T - 1306766329292 $ T - 1306766329292
$47$	$ T + 3351821491776 $ T + 3351821491776
$53$	$ T + 9387813393702 $ T + 9387813393702
$59$	$ T + 28930359275340 $ T + 28930359275340
$61$	$ T - 42393077399702 $ T - 42393077399702
$67$	$ T + 52247243064364 $ T + 52247243064364
$71$	$ T - 27194529024648 $ T - 27194529024648
$73$	$ T + 91604195687878 $ T + 91604195687878
$79$	$ T - 62882111078120 $ T - 62882111078120
$83$	$ T - 223567315949868 $ T - 223567315949868
$89$	$ T + 554198786115210 $ T + 554198786115210
$97$	$ T + 1388870476877374 $ T + 1388870476877374
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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