[N,k,chi] = [162,13,Mod(53,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.53");
S:= CuspForms(chi, 13);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
\(n\)
\(83\)
\(\chi(n)\)
\(-\beta_{1}\)
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.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 941530752 T_{5}^{6} + \cdots + 23\!\cdots\!00 \)
T5^8 - 941530752*T5^6 + 733220309430779904*T5^4 - 144298859497178892809011200*T5^2 + 23488580865196431336747156111360000
acting on \(S_{13}^{\mathrm{new}}(162, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{4} - 2048 T^{2} + 4194304)^{2} \)
(T^4 - 2048*T^2 + 4194304)^2
$3$
\( T^{8} \)
T^8
$5$
\( T^{8} - 941530752 T^{6} + \cdots + 23\!\cdots\!00 \)
T^8 - 941530752*T^6 + 733220309430779904*T^4 - 144298859497178892809011200*T^2 + 23488580865196431336747156111360000
$7$
\( (T^{4} + 135742 T^{3} + \cdots + 14\!\cdots\!25)^{2} \)
(T^4 + 135742*T^3 + 14681667459*T^2 + 508248332718910*T + 14019206660015841025)^2
$11$
\( T^{8} - 3058161293952 T^{6} + \cdots + 47\!\cdots\!00 \)
T^8 - 3058161293952*T^6 + 7184359327535182003912704*T^4 - 6630066688729924129074394364588851200*T^2 + 4700185723131656524329052652746328874795663360000
$13$
\( (T^{4} + 348142 T^{3} + \cdots + 14\!\cdots\!25)^{2} \)
(T^4 + 348142*T^3 + 38479113731379*T^2 - 13353999809311668530*T + 1471329327017800454319016225)^2
$17$
\( (T^{4} + 886383248261760 T^{2} + \cdots + 57\!\cdots\!36)^{2} \)
(T^4 + 886383248261760*T^2 + 5703370348840334994951438336)^2
$19$
\( (T^{2} + 43938482 T - 71954800437263)^{4} \)
(T^2 + 43938482*T - 71954800437263)^4
$23$
\( T^{8} + \cdots + 23\!\cdots\!76 \)
T^8 - 25671326675333760*T^6 + 505007108624971136790202205159424*T^4 - 3953638573423248252456983112233591874052172021760*T^2 + 23719050729295342064299406411356998626960753486465685801167486976
$29$
\( T^{8} + \cdots + 29\!\cdots\!16 \)
T^8 - 973269838526814720*T^6 + 774914746991901004876280309885435904*T^4 - 167732770759403438072518793885128821126945033022341120*T^2 + 29700879682181262591465716956497556293178798821060008378927439220310016
$31$
\( (T^{4} - 1191267068 T^{3} + \cdots + 11\!\cdots\!00)^{2} \)
(T^4 - 1191267068*T^3 + 1080251266808581644*T^2 - 403679859201184234925638640*T + 114830139180663824289349360515600400)^2
$37$
\( (T^{2} + 286643570 T - 18\!\cdots\!75)^{4} \)
(T^2 + 286643570*T - 18054533959259615375)^4
$41$
\( T^{8} + \cdots + 86\!\cdots\!00 \)
T^8 - 35238509839610546688*T^6 + 1148737717978755645380859232529599954944*T^4 - 3277704979615465015938953663000821547910335723698598707200*T^2 + 8651763759941068704304497549770767602373196107450928258802191680520847360000
$43$
\( (T^{4} - 7558366172 T^{3} + \cdots + 26\!\cdots\!04)^{2} \)
(T^4 - 7558366172*T^3 + 108221333898861558732*T^2 + 386175330148321391697739693456*T + 2610436884475813812938462636953602021904)^2
$47$
\( T^{8} + \cdots + 13\!\cdots\!00 \)
T^8 - 92900761162630351488*T^6 + 7480654044366475264331365922648154783744*T^4 - 106826341882245579652241107334809916656677905839968957235200*T^2 + 1322263985058926235924392515871106514041264992106763846830042966866956124160000
$53$
\( (T^{4} + \cdots + 14\!\cdots\!24)^{2} \)
(T^4 + 861442665231158401536*T^2 + 141960549328256946436676640844384828391424)^2
$59$
\( T^{8} + \cdots + 25\!\cdots\!76 \)
T^8 - 7993083567408050024064*T^6 + 47952605382097940154908772697679991841308672*T^4 - 127384010606389255944008855307944037480287786945688961574435291136*T^2 + 253980941898449282473386585255333832536146480467355342719933807005838646579735115595776
$61$
\( (T^{4} + 29181433198 T^{3} + \cdots + 39\!\cdots\!25)^{2} \)
(T^4 + 29181433198*T^3 + 2832551231469951677619*T^2 - 57808278743615774114410708437170*T + 3924341934802352835744205697658039491272225)^2
$67$
\( (T^{4} + 154487577550 T^{3} + \cdots + 31\!\cdots\!21)^{2} \)
(T^4 + 154487577550*T^3 + 18266727826709906265939*T^2 + 865081583849207761050059957805550*T + 31356458554230818265614091738884401076106721)^2
$71$
\( (T^{4} + \cdots + 58\!\cdots\!84)^{2} \)
(T^4 + 38636732258202804627456*T^2 + 58746911673549337014993797342906921080848384)^2
$73$
\( (T^{2} + 89435351714 T - 41\!\cdots\!27)^{4} \)
(T^2 + 89435351714*T - 41336077279200776600927)^4
$79$
\( (T^{4} - 452549584418 T^{3} + \cdots + 24\!\cdots\!89)^{2} \)
(T^4 - 452549584418*T^3 + 155128930228747404326307*T^2 - 22479131714926886116186688726798306*T + 2467327068194105600979234159692322005980221889)^2
$83$
\( T^{8} + \cdots + 40\!\cdots\!16 \)
T^8 - 342142305595309920545280*T^6 + 115053012140001611052315663460276651086562197504*T^4 - 687139835971369478950326993533456331798869764118220386162363470970880*T^2 + 4033450193620779061502830221895502247980489646938186202134269350364027459944113101200162816
$89$
\( (T^{4} + \cdots + 60\!\cdots\!44)^{2} \)
(T^4 + 917942268793340942941824*T^2 + 60630384733403515960490033469430429217477689344)^2
$97$
\( (T^{4} - 2835640618178 T^{3} + \cdots + 40\!\cdots\!25)^{2} \)
(T^4 - 2835640618178*T^3 + 6030902078066864857801539*T^2 - 5699511846130406134741961535443999810*T + 4039921664292102214584743855858765629119253041025)^2
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