LMFDB - Newform orbit 64.22.a.a

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 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 - 128844 q^{3} - 21640950 q^{5} + 768078808 q^{7} + 6140423133 q^{9}+O(q^{10}) $

$ q - 128844 q^{3} - 21640950 q^{5} + 768078808 q^{7} + 6140423133 q^{9} - 94724929188 q^{11} + 80621789794 q^{13} + 2788306561800 q^{15} + 3052282930002 q^{17} - 7920788351740 q^{19} - 98962345937952 q^{21} + 73845437470344 q^{23} - 8506441300625 q^{25} + 556597069939080 q^{27} + 42\!\cdots\!10 q^{29}+ \cdots - 58\!\cdots\!04 q^{99}+O(q^{100}) $

q - 128844 * q^3 - 21640950 * q^5 + 768078808 * q^7 + 6140423133 * q^9 - 94724929188 * q^11 + 80621789794 * q^13 + 2788306561800 * q^15 + 3052282930002 * q^17 - 7920788351740 * q^19 - 98962345937952 * q^21 + 73845437470344 * q^23 - 8506441300625 * q^25 + 556597069939080 * q^27 + 4253031736469010 * q^29 - 1900541176310432 * q^31 + 12204738776298672 * q^33 - 16621955079987600 * q^35 - 22191429912035222 * q^37 - 10387633884218136 * q^39 - 20622803144546358 * q^41 - 193605854685795844 * q^43 - 132884590000096350 * q^45 - 146960504315611632 * q^47 + 31399191215416857 * q^49 - 393268341833177688 * q^51 - 2038267110310687206 * q^53 + 2049937456311048600 * q^55 + 1020546054391588560 * q^57 - 5975882742742352820 * q^59 - 6190617154478149262 * q^61 + 4716328880610265464 * q^63 - 1744732121842464300 * q^65 + 16961315295446680052 * q^67 - 9514541545429002336 * q^69 + 5632758963952293528 * q^71 - 43284759511102937494 * q^73 + 1096003922937727500 * q^75 - 72756210698603447904 * q^77 + 51264938664949064560 * q^79 - 135945187666282668519 * q^81 + 48911854702961049156 * q^83 - 66054302274026781900 * q^85 - 547977621053613124440 * q^87 - 504303489899844009030 * q^89 + 61923888203802085552 * q^91 + 244873327320541300608 * q^93 + 171413384680587753000 * q^95 + 808275058155029184482 * q^97 - 581651146457782106004 * 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

−128844.

−2.16410e7

7.68079e8

6.14042e9

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.a		1
4.b	odd	2	1		64.22.a.g		1
8.b	even	2	1		16.22.a.c		1
8.d	odd	2	1		1.22.a.a	✓	1
24.f	even	2	1		9.22.a.c		1
40.e	odd	2	1		25.22.a.a		1
40.k	even	4	2		25.22.b.a		2
56.e	even	2	1		49.22.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.22.a.a	✓	1	8.d	odd	2	1
9.22.a.c		1	24.f	even	2	1
16.22.a.c		1	8.b	even	2	1
25.22.a.a		1	40.e	odd	2	1
25.22.b.a		2	40.k	even	4	2
49.22.a.a		1	56.e	even	2	1
64.22.a.a		1	1.a	even	1	1	trivial
64.22.a.g		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 128844 $ T3 + 128844 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(64))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 128844 $ T + 128844
$5$	$ 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
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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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