[N,k,chi] = [39,4,Mod(4,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.4");
S:= CuspForms(chi, 4);
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 \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
\(n\)
\(14\)
\(28\)
\(\chi(n)\)
\(1\)
\(1 - \beta_{2}\)
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
\( T_{2}^{10} - 35T_{2}^{8} + 1015T_{2}^{6} - 288T_{2}^{5} - 7350T_{2}^{4} + 44100T_{2}^{2} + 60480T_{2} + 27648 \)
T2^10 - 35*T2^8 + 1015*T2^6 - 288*T2^5 - 7350*T2^4 + 44100*T2^2 + 60480*T2 + 27648
acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} - 35 T^{8} + 1015 T^{6} + \cdots + 27648 \)
T^10 - 35*T^8 + 1015*T^6 - 288*T^5 - 7350*T^4 + 44100*T^2 + 60480*T + 27648
$3$
\( (T^{2} + 3 T + 9)^{5} \)
(T^2 + 3*T + 9)^5
$5$
\( T^{10} + 1105 T^{8} + \cdots + 31632011568 \)
T^10 + 1105*T^8 + 441955*T^6 + 76029795*T^4 + 4880780280*T^2 + 31632011568
$7$
\( T^{10} - 30 T^{9} + \cdots + 14692478786352 \)
T^10 - 30*T^9 - 795*T^8 + 32850*T^7 + 643455*T^6 - 31795416*T^5 + 120275010*T^4 + 7180407000*T^3 - 35828397900*T^2 - 1358157413880*T + 14692478786352
$11$
\( T^{10} - 60 T^{9} + \cdots + 50\!\cdots\!32 \)
T^10 - 60*T^9 - 3590*T^8 + 287400*T^7 + 15404860*T^6 - 1242026664*T^5 + 15813974880*T^4 + 523951682400*T^3 - 8348097093120*T^2 - 226833869197440*T + 5076736586687232
$13$
\( T^{10} - 25 T^{9} + \cdots + 51\!\cdots\!57 \)
T^10 - 25*T^9 + 2625*T^8 - 113660*T^7 - 355615*T^6 - 217453821*T^5 - 781286155*T^4 - 548615110940*T^3 + 27836810854125*T^2 - 582452128062025*T + 51185893014090757
$17$
\( T^{10} + \cdots + 332127005819904 \)
T^10 - 105*T^9 + 11580*T^8 - 455715*T^7 + 32722740*T^6 - 1264758273*T^5 + 64946203785*T^4 - 1375315498080*T^3 + 24803939115360*T^2 - 98962605204480*T + 332127005819904
$19$
\( T^{10} - 180 T^{9} + \cdots + 19\!\cdots\!68 \)
T^10 - 180*T^9 - 4470*T^8 + 2748600*T^7 + 79965180*T^6 - 21017296152*T^5 - 584471890080*T^4 + 97590885328800*T^3 + 7412241749736960*T^2 + 194687487181376640*T + 1965893985956659968
$23$
\( T^{10} + 60 T^{9} + \cdots + 66\!\cdots\!04 \)
T^10 + 60*T^9 + 37410*T^8 - 413280*T^7 + 916714620*T^6 - 13830016248*T^5 + 10434588065280*T^4 - 773345549662560*T^3 + 68963219178606720*T^2 - 2241126885190227840*T + 66482354820461877504
$29$
\( T^{10} + 495 T^{9} + \cdots + 18\!\cdots\!96 \)
T^10 + 495*T^9 + 175110*T^8 + 31525875*T^7 + 4308812370*T^6 + 288990529839*T^5 + 15416711665245*T^4 - 223002180825120*T^3 + 34335830182722300*T^2 + 78475061806506720*T + 182865566122042896
$31$
\( T^{10} + 116790 T^{8} + \cdots + 35\!\cdots\!00 \)
T^10 + 116790*T^8 + 4521262905*T^6 + 65631012933600*T^4 + 334330914729960000*T^2 + 356081403867316531200
$37$
\( T^{10} + 405 T^{9} + \cdots + 48\!\cdots\!32 \)
T^10 + 405*T^9 - 36630*T^8 - 36978525*T^7 + 2395522710*T^6 + 3827547577989*T^5 + 683231529008115*T^4 + 13138032488338800*T^3 - 5810831888207031360*T^2 - 109186407055873704960*T + 48966608306945859858432
$41$
\( T^{10} - 1065 T^{9} + \cdots + 36\!\cdots\!52 \)
T^10 - 1065*T^9 + 428410*T^8 - 53606775*T^7 - 9015747650*T^6 + 2087024967159*T^5 + 431831132193555*T^4 - 143359617874725000*T^3 + 12049145691340424400*T^2 - 112748018174710128000*T + 365794058718280311552
$43$
\( T^{10} + 370 T^{9} + \cdots + 51\!\cdots\!16 \)
T^10 + 370*T^9 + 248505*T^8 + 5784270*T^7 + 15326122815*T^6 - 1921116448896*T^5 + 1289861362081170*T^4 - 188186857631549760*T^3 + 28742867633314016340*T^2 - 1327376643503015066360*T + 51678581902251840345616
$47$
\( T^{10} + 181660 T^{8} + \cdots + 21\!\cdots\!28 \)
T^10 + 181660*T^8 + 7629752500*T^6 + 74704590880080*T^4 + 246476647292518080*T^2 + 215929971648138486528
$53$
\( (T^{5} - 165 T^{4} + \cdots - 46733997168)^{2} \)
(T^5 - 165*T^4 - 175365*T^3 + 54572265*T^2 - 3872088360*T - 46733997168)^2
$59$
\( T^{10} - 780 T^{9} + \cdots + 41\!\cdots\!68 \)
T^10 - 780*T^9 - 49700*T^8 + 196950000*T^7 + 11343554320*T^6 - 38431229739648*T^5 + 4936694147988480*T^4 + 1830518123065344000*T^3 + 78055533491532595200*T^2 - 13095105686249259663360*T + 418560730223463490387968
$61$
\( T^{10} + 1375 T^{9} + \cdots + 87\!\cdots\!25 \)
T^10 + 1375*T^9 + 1173465*T^8 + 630125000*T^7 + 248605267865*T^6 + 70872061087695*T^5 + 15345856775068025*T^4 + 2395364510338982600*T^3 + 274109344746827055225*T^2 + 19595764032602779364575*T + 872077706907594470013025
$67$
\( T^{10} - 1590 T^{9} + \cdots + 13\!\cdots\!88 \)
T^10 - 1590*T^9 + 796245*T^8 + 73863450*T^7 - 159137337465*T^6 - 16264739854908*T^5 + 52917078139025490*T^4 - 18768038265861458400*T^3 + 1998237181890682197540*T^2 + 97436857399426228099320*T + 1395894322140655338839088
$71$
\( T^{10} - 1620 T^{9} + \cdots + 71\!\cdots\!00 \)
T^10 - 1620*T^9 + 847090*T^8 + 44890200*T^7 - 91268403140*T^6 - 3292527811320*T^5 + 9067508567522400*T^4 - 127589959724652000*T^3 - 83545239413320934400*T^2 + 1090451203684625116800*T + 718734914655152629420800
$73$
\( T^{10} + 600615 T^{8} + \cdots + 20\!\cdots\!75 \)
T^10 + 600615*T^8 + 28896798330*T^6 + 327714497099550*T^4 + 1398166335261868725*T^2 + 2045730256408861512075
$79$
\( (T^{5} - 550 T^{4} + \cdots - 920208867136)^{2} \)
(T^5 - 550*T^4 - 641375*T^3 + 74745260*T^2 + 51314196760*T - 920208867136)^2
$83$
\( T^{10} + 3406900 T^{8} + \cdots + 16\!\cdots\!68 \)
T^10 + 3406900*T^8 + 3160358318500*T^6 + 1008935154891789360*T^4 + 99760388286685114967040*T^2 + 1616891650061173003439751168
$89$
\( T^{10} - 2040 T^{9} + \cdots + 28\!\cdots\!28 \)
T^10 - 2040*T^9 + 1320520*T^8 + 136027200*T^7 - 306147533120*T^6 - 19431083228928*T^5 + 54258832705597440*T^4 + 7705830035842560000*T^3 - 1009494609359317401600*T^2 - 165955591497040882237440*T + 28643408919355108985929728
$97$
\( T^{10} + 3750 T^{9} + \cdots + 15\!\cdots\!68 \)
T^10 + 3750*T^9 + 4753035*T^8 + 245756250*T^7 - 3567154531275*T^6 - 885718637150148*T^5 + 2203451121412158300*T^4 + 686867277058961850000*T^3 - 644209092107417455976880*T^2 - 155237932026652815047160000*T + 157149586619843313073725082368
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