# Properties

 Label 16.22.a.b Level $16$ Weight $22$ Character orbit 16.a Self dual yes Analytic conductor $44.716$ Analytic rank $0$ 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] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, 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("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$44.7163750859$$ 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 $$q - 59316 q^{3} + 4975350 q^{5} - 1427425832 q^{7} - 6941965347 q^{9} + 106767894948 q^{11} - 150150565474 q^{13} - 295117860600 q^{15} - 11203980739758 q^{17} - 11024055955460 q^{19} + 84669190650912 q^{21} - 129502845739896 q^{23} - 452083050580625 q^{25} + 10\!\cdots\!00 q^{27}+ \cdots - 74\!\cdots\!56 q^{99}+O(q^{100})$$ 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

## 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 −59316.0 0 4.97535e6 0 −1.42743e9 0 −6.94197e9 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 16.22.a.b 1
4.b odd 2 1 2.22.a.b 1
8.b even 2 1 64.22.a.e 1
8.d odd 2 1 64.22.a.c 1
12.b even 2 1 18.22.a.b 1
20.d odd 2 1 50.22.a.a 1
20.e 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 4.b odd 2 1
16.22.a.b 1 1.a even 1 1 trivial
18.22.a.b 1 12.b even 2 1
50.22.a.a 1 20.d odd 2 1
50.22.b.c 2 20.e even 4 2
64.22.a.c 1 8.d odd 2 1
64.22.a.e 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 59316$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

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