LMFDB - Newform orbit 9.22.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	9.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:	$25.1529609858$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
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 + 288 q^{2} - 2014208 q^{4} - 21640950 q^{5} - 768078808 q^{7} - 1184071680 q^{8}+O(q^{10}) $

$ q + 288 q^{2} - 2014208 q^{4} - 21640950 q^{5} - 768078808 q^{7} - 1184071680 q^{8} - 6232593600 q^{10} + 94724929188 q^{11} - 80621789794 q^{13} - 221206696704 q^{14} + 3883087691776 q^{16} - 3052282930002 q^{17} - 7920788351740 q^{19} + 43589374617600 q^{20} + 27280779606144 q^{22} + 73845437470344 q^{23} - 8506441300625 q^{25} - 23219075460672 q^{26} + 15\!\cdots\!64 q^{28}+ \cdots + 90\!\cdots\!16 q^{98}+O(q^{100}) $

q + 288 * q^2 - 2014208 * q^4 - 21640950 * q^5 - 768078808 * q^7 - 1184071680 * q^8 - 6232593600 * q^10 + 94724929188 * q^11 - 80621789794 * q^13 - 221206696704 * q^14 + 3883087691776 * q^16 - 3052282930002 * q^17 - 7920788351740 * q^19 + 43589374617600 * q^20 + 27280779606144 * q^22 + 73845437470344 * q^23 - 8506441300625 * q^25 - 23219075460672 * q^26 + 1547070479704064 * q^28 + 4253031736469010 * q^29 + 1900541176310432 * q^31 + 3601507547086848 * q^32 - 879057483840576 * q^34 + 16621955079987600 * q^35 + 22191429912035222 * q^37 - 2281187045301120 * q^38 + 25624436023296000 * q^40 + 20622803144546358 * q^41 - 193605854685795844 * q^43 - 190795710169903104 * q^44 + 21267485991459072 * q^46 - 146960504315611632 * q^47 + 31399191215416857 * q^49 - 2449855094580000 * q^50 + 162389053977393152 * q^52 - 2038267110310687206 * q^53 - 2049937456311048600 * q^55 + 909460364560957440 * q^56 + 1224873140103074880 * q^58 + 5975882742742352820 * q^59 + 6190617154478149262 * q^61 + 547355858777404416 * q^62 - 7106190945422409728 * q^64 + 1744732121842464300 * q^65 + 16961315295446680052 * q^67 + 6147932695873468416 * q^68 + 4787123063036428800 * q^70 + 5632758963952293528 * q^71 - 43284759511102937494 * q^73 + 6391131814666143936 * q^74 + 15954115264381521920 * q^76 - 72756210698603447904 * q^77 - 51264938664949064560 * q^79 - 84033706583339827200 * q^80 + 5939367305629351104 * q^82 - 48911854702961049156 * q^83 + 66054302274026781900 * q^85 - 55758486149509203072 * q^86 - 112161106041516195840 * q^88 + 504303489899844009030 * q^89 + 61923888203802085552 * q^91 - 148740070916266647552 * q^92 - 42324625242896150016 * q^94 + 171413384680587753000 * q^95 + 808275058155029184482 * q^97 + 9042967070040054816 * q^98

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

288.000

−2.01421e6

−2.16410e7

−7.68079e8

−1.18407e9

−6.23259e9

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$

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	9.22.a.c		1
3.b	odd	2	1		1.22.a.a	✓	1
12.b	even	2	1		16.22.a.c		1
15.d	odd	2	1		25.22.a.a		1
15.e	even	4	2		25.22.b.a		2
21.c	even	2	1		49.22.a.a		1
24.f	even	2	1		64.22.a.a		1
24.h	odd	2	1		64.22.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.22.a.a	✓	1	3.b	odd	2	1
9.22.a.c		1	1.a	even	1	1	trivial
16.22.a.c		1	12.b	even	2	1
25.22.a.a		1	15.d	odd	2	1
25.22.b.a		2	15.e	even	4	2
49.22.a.a		1	21.c	even	2	1
64.22.a.a		1	24.f	even	2	1
64.22.a.g		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 288 $ T2 - 288 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(9))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 288 $ T - 288
$3$	$ T $ T
$5$	$ T + 21640950 $ T + 21640950
$7$	$ T + 768078808 $ T + 768078808
$11$	$ T - 94724929188 $ T - 94724929188
$13$	$ T + 80621789794 $ T + 80621789794
$17$	$ T + 3052282930002 $ T + 3052282930002
$19$	$ T + 7920788351740 $ T + 7920788351740
$23$	$ T - 73845437470344 $ T - 73845437470344
$29$	$ T - 4253031736469010 $ T - 4253031736469010
$31$	$ T - 1900541176310432 $ T - 1900541176310432
$37$	$ T - 22\!\cdots\!22 $ T - 22191429912035222
$41$	$ T - 20\!\cdots\!58 $ T - 20622803144546358
$43$	$ T + 19\!\cdots\!44 $ T + 193605854685795844
$47$	$ T + 14\!\cdots\!32 $ T + 146960504315611632
$53$	$ T + 20\!\cdots\!06 $ T + 2038267110310687206
$59$	$ T - 59\!\cdots\!20 $ T - 5975882742742352820
$61$	$ T - 61\!\cdots\!62 $ T - 6190617154478149262
$67$	$ T - 16\!\cdots\!52 $ T - 16961315295446680052
$71$	$ T - 56\!\cdots\!28 $ T - 5632758963952293528
$73$	$ T + 43\!\cdots\!94 $ T + 43284759511102937494
$79$	$ T + 51\!\cdots\!60 $ T + 51264938664949064560
$83$	$ T + 48\!\cdots\!56 $ T + 48911854702961049156
$89$	$ T - 50\!\cdots\!30 $ T - 504303489899844009030
$97$	$ T - 80\!\cdots\!82 $ T - 808275058155029184482
show more
show less

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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more