# Properties

 Label 64.22.a.g Level $64$ Weight $22$ Character orbit 64.a Self dual yes Analytic conductor $178.866$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

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: $$1$$ 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 $$q + 128844 q^{3} - 21640950 q^{5} - 768078808 q^{7} + 6140423133 q^{9} + 94724929188 q^{11} + 80621789794 q^{13} - 2788306561800 q^{15} + 3052282930002 q^{17} + 7920788351740 q^{19} - 98962345937952 q^{21} - 73845437470344 q^{23} - 8506441300625 q^{25} - 556597069939080 q^{27} + 42\!\cdots\!10 q^{29}+ \cdots + 58\!\cdots\!04 q^{99}+O(q^{100})$$ 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

## 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
 0
0 128844. 0 −2.16410e7 0 −7.68079e8 0 6.14042e9 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.g 1
4.b odd 2 1 64.22.a.a 1
8.b even 2 1 1.22.a.a 1
8.d odd 2 1 16.22.a.c 1
24.h odd 2 1 9.22.a.c 1
40.f even 2 1 25.22.a.a 1
40.i odd 4 2 25.22.b.a 2
56.h odd 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.b even 2 1
9.22.a.c 1 24.h odd 2 1
16.22.a.c 1 8.d odd 2 1
25.22.a.a 1 40.f even 2 1
25.22.b.a 2 40.i odd 4 2
49.22.a.a 1 56.h odd 2 1
64.22.a.a 1 4.b odd 2 1
64.22.a.g 1 1.a even 1 1 trivial

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 128844$$
$5$ $$T + 21640950$$
$7$ $$T + 768078808$$
$11$ $$T - 94724929188$$
$13$ $$T - 80621789794$$
$17$ $$T - 3052282930002$$
$19$ $$T - 7920788351740$$
$23$ $$T + 73845437470344$$
$29$ $$T - 4253031736469010$$
$31$ $$T - 1900541176310432$$
$37$ $$T + 22\!\cdots\!22$$
$41$ $$T + 20\!\cdots\!58$$
$43$ $$T - 19\!\cdots\!44$$
$47$ $$T - 14\!\cdots\!32$$
$53$ $$T + 20\!\cdots\!06$$
$59$ $$T - 59\!\cdots\!20$$
$61$ $$T + 61\!\cdots\!62$$
$67$ $$T + 16\!\cdots\!52$$
$71$ $$T + 56\!\cdots\!28$$
$73$ $$T + 43\!\cdots\!94$$
$79$ $$T + 51\!\cdots\!60$$
$83$ $$T + 48\!\cdots\!56$$
$89$ $$T + 50\!\cdots\!30$$
$97$ $$T - 80\!\cdots\!82$$