Properties

 Label 2.22.a.b Level $2$ Weight $22$ Character orbit 2.a Self dual yes Analytic conductor $5.590$ 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] = [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)

 $$f(q)$$ $$=$$ $$q + 1024 q^{2} + 59316 q^{3} + 1048576 q^{4} + 4975350 q^{5} + 60739584 q^{6} + 1427425832 q^{7} + 1073741824 q^{8} - 6941965347 q^{9}+O(q^{10})$$ 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 $$q + 1024 q^{2} + 59316 q^{3} + 1048576 q^{4} + 4975350 q^{5} + 60739584 q^{6} + 1427425832 q^{7} + 1073741824 q^{8} - 6941965347 q^{9} + 5094758400 q^{10} - 106767894948 q^{11} + 62197334016 q^{12} - 150150565474 q^{13} + 1461684051968 q^{14} + 295117860600 q^{15} + 1099511627776 q^{16} - 11203980739758 q^{17} - 7108572515328 q^{18} + 11024055955460 q^{19} + 5217032601600 q^{20} + 84669190650912 q^{21} - 109330324426752 q^{22} + 129502845739896 q^{23} + 63690070032384 q^{24} - 452083050580625 q^{25} - 153754179045376 q^{26} - 10\!\cdots\!00 q^{27}+ \cdots + 74\!\cdots\!56 q^{99}+O(q^{100})$$ 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
 0
1024.00 59316.0 1.04858e6 4.97535e6 6.07396e7 1.42743e9 1.07374e9 −6.94197e9 5.09476e9
 $$n$$: 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 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 $$T_{3} - 59316$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(2))$$.

Hecke characteristic polynomials

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