[N,k,chi] = [78,18,Mod(1,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.1");
S:= CuspForms(chi, 18);
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
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
\( p \)
Sign
\(2\)
\(1\)
\(3\)
\(1\)
\(13\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4540T_{5}^{3} - 1600115250936T_{5}^{2} - 686087389157576800T_{5} - 48609776967062726960000 \)
T5^4 - 4540*T5^3 - 1600115250936*T5^2 - 686087389157576800*T5 - 48609776967062726960000
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(78))\).
$p$
$F_p(T)$
$2$
\( (T + 256)^{4} \)
(T + 256)^4
$3$
\( (T + 6561)^{4} \)
(T + 6561)^4
$5$
\( T^{4} - 4540 T^{3} + \cdots - 48\!\cdots\!00 \)
T^4 - 4540*T^3 - 1600115250936*T^2 - 686087389157576800*T - 48609776967062726960000
$7$
\( T^{4} + 31248260 T^{3} + \cdots - 73\!\cdots\!40 \)
T^4 + 31248260*T^3 - 374773988045352*T^2 - 16006376395726556306656*T - 73641920379278178776892659840
$11$
\( T^{4} - 424508888 T^{3} + \cdots + 30\!\cdots\!00 \)
T^4 - 424508888*T^3 - 1084124688363616584*T^2 + 251673902661137362055906080*T + 303770022675385797408185631638035600
$13$
\( (T + 815730721)^{4} \)
(T + 815730721)^4
$17$
\( T^{4} + 8600960928 T^{3} + \cdots + 11\!\cdots\!40 \)
T^4 + 8600960928*T^3 - 2488452395285937299832*T^2 + 3762456486161092825035507483456*T + 115099555701460274318010306088078270773840
$19$
\( T^{4} - 63345491892 T^{3} + \cdots - 96\!\cdots\!60 \)
T^4 - 63345491892*T^3 + 158778176755173570936*T^2 + 15296725143680737087482885939168*T - 96877889685642560053325657941806858264960
$23$
\( T^{4} + 346898449760 T^{3} + \cdots - 10\!\cdots\!88 \)
T^4 + 346898449760*T^3 - 276263661422996468600736*T^2 - 127889159849489631415034412890240512*T - 10585289083013459953651077007088111820541445888
$29$
\( T^{4} - 2796914407528 T^{3} + \cdots - 40\!\cdots\!00 \)
T^4 - 2796914407528*T^3 - 17644974681128631176525160*T^2 + 58182624973467243304008900963605103200*T - 40841447826723145030941373547249857304830571990000
$31$
\( T^{4} - 4046594993164 T^{3} + \cdots + 68\!\cdots\!20 \)
T^4 - 4046594993164*T^3 - 15787754901498528614386776*T^2 + 31088747363951134850543374805805370784*T + 68589223369871631855363577865483996756973784270720
$37$
\( T^{4} + 33842172103056 T^{3} + \cdots - 22\!\cdots\!68 \)
T^4 + 33842172103056*T^3 - 441022699445153791522594968*T^2 - 25667721277533977905985373442173606045696*T - 225091870634861126656155884490712570691738592291928368
$41$
\( T^{4} + 139766232154884 T^{3} + \cdots + 57\!\cdots\!00 \)
T^4 + 139766232154884*T^3 + 3343215670366732654384861128*T^2 - 107889814636150260593006239922617505476320*T + 571177946975899533176624169429004448716103846475792000
$43$
\( T^{4} + 120980001751736 T^{3} + \cdots + 36\!\cdots\!60 \)
T^4 + 120980001751736*T^3 + 4671641397748960981606341984*T^2 + 71457629940323837267830864564984887737984*T + 365235799881408348859973032063922803148961712539900160
$47$
\( T^{4} + 275562239626048 T^{3} + \cdots - 17\!\cdots\!00 \)
T^4 + 275562239626048*T^3 - 77048446812562983741143646360*T^2 - 27447388562159985652038764456840609621892800*T - 1705117851359508796874969579901976551660403254148108094000
$53$
\( T^{4} + \cdots - 60\!\cdots\!56 \)
T^4 - 1073308288315752*T^3 + 367534592247537620888991864120*T^2 - 38650756920841746057635580784732148829139488*T - 602499368571065370004517930149401870062397767009557382256
$59$
\( T^{4} + \cdots - 16\!\cdots\!00 \)
T^4 - 4054988606581312*T^3 + 4469681854289354419313159822952*T^2 + 171113543127380067478129839081274917893522240*T - 1679327972161876885645530574870009884044497400161628748734000
$61$
\( T^{4} - 150096617284408 T^{3} + \cdots + 28\!\cdots\!40 \)
T^4 - 150096617284408*T^3 - 1000204095475231578998135730696*T^2 + 267690691097903818448309241194244043564123808*T + 28629038855194673939075427936086325354641263119668756509840
$67$
\( T^{4} + \cdots - 12\!\cdots\!68 \)
T^4 - 6120599470256860*T^3 + 2449631152489286806235908656936*T^2 + 5902591990143688343544520356139192680373715744*T - 1206052320955081017903469687594512807560176439280323102154368
$71$
\( T^{4} + \cdots - 10\!\cdots\!00 \)
T^4 - 17451507704387576*T^3 + 75899657574652812976849838020440*T^2 + 126098002664332487613193151969662174707712007200*T - 1016563034434743784312850075217908793427291666622012874646550000
$73$
\( T^{4} + \cdots + 33\!\cdots\!60 \)
T^4 - 3526487244453968*T^3 - 132014074994024834987707047884184*T^2 + 254468346453262043885487231205120336076823448192*T + 33104449921848112450917991491436945010255391646444321130545360
$79$
\( T^{4} + \cdots + 56\!\cdots\!80 \)
T^4 - 15892613834559632*T^3 - 477118209498060255427332483406464*T^2 + 3173052340430913856530054073537250149962136695808*T + 56641387469916837999047835778606278425030766419223352482646036480
$83$
\( T^{4} + \cdots + 62\!\cdots\!96 \)
T^4 + 10194030244954032*T^3 - 1104594611789405796360267672941688*T^2 + 5534181823005953558861665637358298313197996234752*T + 62099557630581597027585463086542744694030986780932214312959618896
$89$
\( T^{4} + \cdots + 44\!\cdots\!80 \)
T^4 - 139823337727620084*T^3 + 6426814610656054833291186449566104*T^2 - 108723528694317535446637033938955732723272096231456*T + 442504437377423460061783655335570617351544566029631582653915153280
$97$
\( T^{4} + \cdots - 12\!\cdots\!04 \)
T^4 - 35860734633045344*T^3 - 5850415977532984851991127230825272*T^2 - 79607362611356056373694852058363667333329758499136*T - 126324538893578098977310805993015487459045336060165771057061091504
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