LMFDB - Newform orbit 64.22.a.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("64.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.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:	$178.865500344$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
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 + 59316 q^{3} - 4975350 q^{5} - 1427425832 q^{7} - 6941965347 q^{9}+O(q^{10}) $

$ q + 59316 q^{3} - 4975350 q^{5} - 1427425832 q^{7} - 6941965347 q^{9} - 106767894948 q^{11} + 150150565474 q^{13} - 295117860600 q^{15} - 11203980739758 q^{17} + 11024055955460 q^{19} - 84669190650912 q^{21} - 129502845739896 q^{23} - 452083050580625 q^{25} - 10\!\cdots\!00 q^{27}+ \cdots + 74\!\cdots\!56 q^{99}+O(q^{100}) $

q + 59316 * q^3 - 4975350 * q^5 - 1427425832 * q^7 - 6941965347 * q^9 - 106767894948 * q^11 + 150150565474 * q^13 - 295117860600 * q^15 - 11203980739758 * q^17 + 11024055955460 * q^19 - 84669190650912 * q^21 - 129502845739896 * q^23 - 452083050580625 * q^25 - 1032235927111800 * q^27 - 2382370826608110 * q^29 + 878552957377888 * q^31 - 6333044456735568 * q^33 + 7101943113241200 * q^35 - 31130005856560022 * q^37 + 8906330941655784 * q^39 - 24612925945718838 * q^41 - 133386119963316484 * q^43 + 34538707289196450 * q^45 + 192524017446421008 * q^47 + 1478998641777608217 * q^49 - 664575321559485528 * q^51 + 594166360130841114 * q^53 + 531207646129531800 * q^55 + 653902903054065360 * q^57 - 2955954134483673780 * q^59 - 7984150090052846222 * q^61 + 9909140661156643704 * q^63 - 747051615931065900 * q^65 + 4837041486709240052 * q^67 - 7681590797907671136 * q^69 - 8849017338933008232 * q^71 + 36684416180434869866 * q^73 - 26815758228240352500 * q^75 + 152403251277037496736 * q^77 - 33840609578636773520 * q^79 + 11387303200042927641 * q^81 + 204214536301552085316 * q^83 + 55743725573554965300 * q^85 - 141312707951086652760 * q^87 - 41024056743692272710 * q^89 - 214328795846994924368 * q^91 + 52112247219826804608 * q^93 - 54848536797997911000 * q^95 - 727592440524100077598 * q^97 + 741179026901152366956 * 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

59316.0

−4.97535e6

−1.42743e9

−6.94197e9

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	64.22.a.e		1
4.b	odd	2	1		64.22.a.c		1
8.b	even	2	1		16.22.a.b		1
8.d	odd	2	1		2.22.a.b	✓	1
24.f	even	2	1		18.22.a.b		1
40.e	odd	2	1		50.22.a.a		1
40.k	even	4	2		50.22.b.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.22.a.b	✓	1	8.d	odd	2	1
16.22.a.b		1	8.b	even	2	1
18.22.a.b		1	24.f	even	2	1
50.22.a.a		1	40.e	odd	2	1
50.22.b.c		2	40.k	even	4	2
64.22.a.c		1	4.b	odd	2	1
64.22.a.e		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 59316 $ T3 - 59316 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(64))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ 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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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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

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