LMFDB - Newform orbit 2.22.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2,22,Mod(1,2)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2, base_ring=CyclotomicField(1))

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

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

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

chi := DirichletCharacter("2.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$N$	$=$	$2$
Weight:	$k$	$=$	$22$
Character orbit:	$[\chi]$	$=$	2.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:	$5.58954688574$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 + 1024 q^{2} + 59316 q^{3} + 1048576 q^{4} + 4975350 q^{5} + 60739584 q^{6} + 1427425832 q^{7} + 1073741824 q^{8} - 6941965347 q^{9} + 5094758400 q^{10} - 106767894948 q^{11} + 62197334016 q^{12} - 150150565474 q^{13}+ \cdots + 74\!\cdots\!56 q^{99}+O(q^{100})

q + 1024 * q^2 + 59316 * q^3 + 1048576 * q^4 + 4975350 * q^5 + 60739584 * q^6 + 1427425832 * q^7 + 1073741824 * q^8 - 6941965347 * q^9 + 5094758400 * q^10 - 106767894948 * q^11 + 62197334016 * q^12 - 150150565474 * q^13 + 1461684051968 * q^14 + 295117860600 * q^15 + 1099511627776 * q^16 - 11203980739758 * q^17 - 7108572515328 * q^18 + 11024055955460 * q^19 + 5217032601600 * q^20 + 84669190650912 * q^21 - 109330324426752 * q^22 + 129502845739896 * q^23 + 63690070032384 * q^24 - 452083050580625 * q^25 - 153754179045376 * q^26 - 1032235927111800 * q^27 + 1496764469215232 * q^28 + 2382370826608110 * q^29 + 302200689254400 * q^30 - 878552957377888 * q^31 + 1125899906842624 * q^32 - 6333044456735568 * q^33 - 11472876277512192 * q^34 + 7101943113241200 * q^35 - 7279178255695872 * q^36 + 31130005856560022 * q^37 + 11288633298391040 * q^38 - 8906330941655784 * q^39 + 5342241384038400 * q^40 - 24612925945718838 * q^41 + 86701251226533888 * q^42 - 133386119963316484 * q^43 - 111954252212994048 * q^44 - 34538707289196450 * q^45 + 132610914037653504 * q^46 - 192524017446421008 * q^47 + 65218631713161216 * q^48 + 1478998641777608217 * q^49 - 462933043794560000 * q^50 - 664575321559485528 * q^51 - 157444279342465024 * q^52 - 594166360130841114 * q^53 - 1057009589362483200 * q^54 - 531207646129531800 * q^55 + 1532686816476397568 * q^56 + 653902903054065360 * q^57 + 2439547726446704640 * q^58 - 2955954134483673780 * q^59 + 309453505796505600 * q^60 + 7984150090052846222 * q^61 - 899638228354957312 * q^62 - 9909140661156643704 * q^63 + 1152921504606846976 * q^64 - 747051615931065900 * q^65 - 6485037523697221632 * q^66 + 4837041486709240052 * q^67 - 11748225308172484608 * q^68 + 7681590797907671136 * q^69 + 7272389747958988800 * q^70 + 8849017338933008232 * q^71 - 7453878533832572928 * q^72 + 36684416180434869866 * q^73 + 31877125997117462528 * q^74 - 26815758228240352500 * q^75 + 11559560497552424960 * q^76 - 152403251277037496736 * q^77 - 9120082884255522816 * q^78 + 33840609578636773520 * q^79 + 5470455177255321600 * q^80 + 11387303200042927641 * q^81 - 25203636168416090112 * q^82 + 204214536301552085316 * q^83 + 88782081255970701312 * q^84 - 55743725573554965300 * q^85 - 136587386842436079616 * q^86 + 141312707951086652760 * q^87 - 114641154266105905152 * q^88 - 41024056743692272710 * q^89 - 35367636264137164800 * q^90 - 214328795846994924368 * q^91 + 135793575974557188096 * q^92 - 52112247219826804608 * q^93 - 197144593865135112192 * q^94 + 54848536797997911000 * q^95 + 66783878874277085184 * q^96 - 727592440524100077598 * q^97 + 1514494609180270814208 * q^98 + 741179026901152366956 * 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

1024.00

59316.0

1.04858e6

4.97535e6

6.07396e7

1.42743e9

1.07374e9

−6.94197e9

5.09476e9

Atkin-Lehner signs

$p$	Sign
$2$	$-1$

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	2.22.a.b	✓	1
3.b	odd	2	1		18.22.a.b		1
4.b	odd	2	1		16.22.a.b		1
5.b	even	2	1		50.22.a.a		1
5.c	odd	4	2		50.22.b.c		2
8.b	even	2	1		64.22.a.c		1
8.d	odd	2	1		64.22.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.22.a.b	✓	1	1.a	even	1	1	trivial
16.22.a.b		1	4.b	odd	2	1
18.22.a.b		1	3.b	odd	2	1
50.22.a.a		1	5.b	even	2	1
50.22.b.c		2	5.c	odd	4	2
64.22.a.c		1	8.b	even	2	1
64.22.a.e		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 59316$ T3 - 59316 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(2))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 1024$ T - 1024
$3$	$T - 59316$ T - 59316
$5$	$T - 4975350$ T - 4975350
$7$	$T - 1427425832$ T - 1427425832
$11$	$T + 106767894948$ T + 106767894948
$13$	$T + 150150565474$ T + 150150565474
$17$	$T + 11203980739758$ T + 11203980739758
$19$	$T - 11024055955460$ T - 11024055955460
$23$	$T - 129502845739896$ T - 129502845739896
$29$	$T - 2382370826608110$ T - 2382370826608110
$31$	$T + 878552957377888$ T + 878552957377888
$37$	$T - 31\!\cdots\!22$ T - 31130005856560022
$41$	$T + 24\!\cdots\!38$ T + 24612925945718838
$43$	$T + 13\!\cdots\!84$ T + 133386119963316484
$47$	$T + 19\!\cdots\!08$ T + 192524017446421008
$53$	$T + 59\!\cdots\!14$ T + 594166360130841114
$59$	$T + 29\!\cdots\!80$ T + 2955954134483673780
$61$	$T - 79\!\cdots\!22$ T - 7984150090052846222
$67$	$T - 48\!\cdots\!52$ T - 4837041486709240052
$71$	$T - 88\!\cdots\!32$ T - 8849017338933008232
$73$	$T - 36\!\cdots\!66$ T - 36684416180434869866
$79$	$T - 33\!\cdots\!20$ T - 33840609578636773520
$83$	$T - 20\!\cdots\!16$ T - 204214536301552085316
$89$	$T + 41\!\cdots\!10$ T + 41024056743692272710
$97$	$T + 72\!\cdots\!98$ T + 727592440524100077598
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