[N,k,chi] = [50,13,Mod(7,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.7");
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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
\(n\)
\(27\)
\(\chi(n)\)
\(-\beta_{1}\)
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
\( T_{3}^{6} + 296 T_{3}^{5} + 43808 T_{3}^{4} + 108468072 T_{3}^{3} + 1124902056996 T_{3}^{2} + 448013764587024 T_{3} + 89\!\cdots\!28 \)
T3^6 + 296*T3^5 + 43808*T3^4 + 108468072*T3^3 + 1124902056996*T3^2 + 448013764587024*T3 + 89215026326576928
acting on \(S_{13}^{\mathrm{new}}(50, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{2} + 64 T + 2048)^{3} \)
(T^2 + 64*T + 2048)^3
$3$
\( T^{6} + 296 T^{5} + \cdots + 89\!\cdots\!28 \)
T^6 + 296*T^5 + 43808*T^4 + 108468072*T^3 + 1124902056996*T^2 + 448013764587024*T + 89215026326576928
$5$
\( T^{6} \)
T^6
$7$
\( T^{6} - 322104 T^{5} + \cdots + 64\!\cdots\!28 \)
T^6 - 322104*T^5 + 51875493408*T^4 - 1929167353612328*T^3 + 27004156991449227396*T^2 - 18723166251997896629720976*T + 6490796113556805479487011037728
$11$
\( (T^{3} - 826356 T^{2} + \cdots - 16\!\cdots\!08)^{2} \)
(T^3 - 826356*T^2 - 3376440046188*T - 1657708044591642208)^2
$13$
\( T^{6} - 4646814 T^{5} + \cdots + 20\!\cdots\!48 \)
T^6 - 4646814*T^5 + 10796440175298*T^4 + 48303371497979584512*T^3 + 81017553406703064914236356*T^2 + 58303944443233481771863534940424*T + 20979095237393381483691130412978995848
$17$
\( T^{6} + 51200226 T^{5} + \cdots + 20\!\cdots\!68 \)
T^6 + 51200226*T^5 + 1310731571225538*T^4 + 5120091423010355658752*T^3 + 85784497550976565835451226116*T^2 + 5891808761998560742595056738452255624*T + 202329158991313795349312349389211565223919368
$19$
\( T^{6} + \cdots + 34\!\cdots\!00 \)
T^6 + 10260114567502800*T^4 + 17075452128606905532593016960000*T^2 + 3430180178798540971553006502508049305600000000
$23$
\( T^{6} + 105826896 T^{5} + \cdots + 50\!\cdots\!28 \)
T^6 + 105826896*T^5 + 5599665958497408*T^4 + 1372550708793315898451672*T^3 + 6656913508559183095885028497683396*T^2 + 816466783149148259589091486743236101899024*T + 50069601109343581041736110961118118741487804509728
$29$
\( T^{6} + \cdots + 10\!\cdots\!00 \)
T^6 + 1234964924172590400*T^4 + 298000945449402143315530472048640000*T^2 + 10557421309060459766587483779054248903848493056000000
$31$
\( (T^{3} + 1333558584 T^{2} + \cdots - 55\!\cdots\!48)^{2} \)
(T^3 + 1333558584*T^2 - 386224050031963548*T - 55393481249818849452707248)^2
$37$
\( T^{6} + 1747956246 T^{5} + \cdots + 38\!\cdots\!28 \)
T^6 + 1747956246*T^5 + 1527675518965206258*T^4 + 341954289351733667165818272*T^3 + 36327553223800369506256296145877796*T^2 - 1676790270870744107303521895441656006007976*T + 38698251918666490377750653424707074725360887067528
$41$
\( (T^{3} - 11361049116 T^{2} + \cdots - 18\!\cdots\!48)^{2} \)
(T^3 - 11361049116*T^2 + 29613898778711973252*T - 18505141522698436738624870048)^2
$43$
\( T^{6} - 15890524824 T^{5} + \cdots + 78\!\cdots\!68 \)
T^6 - 15890524824*T^5 + 126254389591080115488*T^4 - 415535817172921489153545174648*T^3 + 696881584739014354079282525206179430116*T^2 - 104285693510377694940902871900336891580509433776*T + 7802979809699924793734071454061705595765264610817780768
$47$
\( T^{6} - 18495531264 T^{5} + \cdots + 57\!\cdots\!48 \)
T^6 - 18495531264*T^5 + 171042338368800718848*T^4 - 445062363061175251349697398088*T^3 + 32270119219753852722220842281739508356*T^2 + 1931403628961778544758029690867494135805667367824*T + 57798360653146377569397708372595747024399733019210290684448
$53$
\( T^{6} - 88020413514 T^{5} + \cdots + 11\!\cdots\!48 \)
T^6 - 88020413514*T^5 + 3873796597587776914098*T^4 - 78669964078123347549806096686688*T^3 + 829753909195246183820903270795627203307556*T^2 - 1374159904719292308170723601580160411791867563201576*T + 1137876798658023742803705463914945940775406297713610588378248
$59$
\( T^{6} + \cdots + 19\!\cdots\!00 \)
T^6 + 6777827948732039259600*T^4 + 11976764918152121773459868672347233959040000*T^2 + 1977855845314915225587272358627630552701996884752076509184000000
$61$
\( (T^{3} - 145897445676 T^{2} + \cdots + 10\!\cdots\!12)^{2} \)
(T^3 - 145897445676*T^2 + 5160097467035877245892*T + 10331748489076313306075522970912)^2
$67$
\( T^{6} - 3887251464 T^{5} + \cdots + 11\!\cdots\!48 \)
T^6 - 3887251464*T^5 + 7555361972185071648*T^4 - 1446234337416902757258026881760888*T^3 + 385311919434875689058287469853676672726854756*T^2 - 29886464133693574921691207955099951505515865314739710576*T + 1159061909795809810341594095942072906919310171116457944439745999648
$71$
\( (T^{3} - 464567507736 T^{2} + \cdots - 28\!\cdots\!28)^{2} \)
(T^3 - 464567507736*T^2 + 65312024549185524756132*T - 2894339609972314794594396038041328)^2
$73$
\( T^{6} + 12678070086 T^{5} + \cdots + 81\!\cdots\!48 \)
T^6 + 12678070086*T^5 + 80366730552764023698*T^4 - 225650408066871405181981155909088*T^3 + 59417775238037335472478011235260739424081956*T^2 - 986076658172702442488377168451112907873208479137273576*T + 8182292015290564817595254456679450938784411405505952740098455048
$79$
\( T^{6} + \cdots + 32\!\cdots\!00 \)
T^6 + 235762185819835185792000*T^4 + 7320913492351636578049292649734019549757440000*T^2 + 32903383316959019201135791532989575342610274560280210335334400000000
$83$
\( T^{6} + 68676615456 T^{5} + \cdots + 27\!\cdots\!08 \)
T^6 + 68676615456*T^5 + 2358238755245649043968*T^4 - 39107127401779909170335214109767768*T^3 + 51003889322391246065789018551551230218418020836*T^2 - 5329195065607781800879569420487532367828303739122701960976*T + 278413278130441410630370169433212964132108792006448155409441471288608
$89$
\( T^{6} + \cdots + 33\!\cdots\!00 \)
T^6 + 278500741527109713907200*T^4 + 18913704507504721233187317265005526294241280000*T^2 + 336219098581922711501190456810602591504375728929801452649447424000000
$97$
\( T^{6} + 675735777846 T^{5} + \cdots + 44\!\cdots\!28 \)
T^6 + 675735777846*T^5 + 228309420730569332199858*T^4 - 111808017868662636453555890345864928*T^3 + 2416201371827101918754283700381277559001332677796*T^2 + 1458917819499966490070811986963007884519594994444964678180824*T + 440451948432807072612301682114371340448187928218654378913068065500494728
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