[N,k,chi] = [1859,4,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
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
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
\(11\)
\(-1\)
\(13\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 59T_{2}^{7} + 12T_{2}^{6} + 1144T_{2}^{5} - 345T_{2}^{4} - 7888T_{2}^{3} + 2245T_{2}^{2} + 9710T_{2} + 2988 \)
T2^9 - 59*T2^7 + 12*T2^6 + 1144*T2^5 - 345*T2^4 - 7888*T2^3 + 2245*T2^2 + 9710*T2 + 2988
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
$p$
$F_p(T)$
$2$
\( T^{9} - 59 T^{7} + 12 T^{6} + \cdots + 2988 \)
T^9 - 59*T^7 + 12*T^6 + 1144*T^5 - 345*T^4 - 7888*T^3 + 2245*T^2 + 9710*T + 2988
$3$
\( T^{9} - 8 T^{8} - 135 T^{7} + \cdots + 283048 \)
T^9 - 8*T^8 - 135*T^7 + 1015*T^6 + 4859*T^5 - 27709*T^4 - 83114*T^3 + 145675*T^2 + 475226*T + 283048
$5$
\( T^{9} + 30 T^{8} + \cdots + 5425892224 \)
T^9 + 30*T^8 - 421*T^7 - 17070*T^6 + 38148*T^5 + 3171776*T^4 - 484320*T^3 - 238114176*T^2 + 66954624*T + 5425892224
$7$
\( T^{9} + 25 T^{8} + \cdots + 18338418984 \)
T^9 + 25*T^8 - 1345*T^7 - 36052*T^6 + 462108*T^5 + 14267352*T^4 - 24653381*T^3 - 1339235989*T^2 + 1196462545*T + 18338418984
$11$
\( (T - 11)^{9} \)
(T - 11)^9
$13$
\( T^{9} \)
T^9
$17$
\( T^{9} - 53 T^{8} + \cdots + 63\!\cdots\!76 \)
T^9 - 53*T^8 - 27946*T^7 + 1129936*T^6 + 265532384*T^5 - 7249591648*T^4 - 963969465568*T^3 + 12088009420992*T^2 + 1011834889297664*T + 6386620512459776
$19$
\( T^{9} + 69 T^{8} + \cdots - 14\!\cdots\!00 \)
T^9 + 69*T^8 - 22970*T^7 - 1857827*T^6 + 135681434*T^5 + 13489287813*T^4 - 85405045419*T^3 - 24839285116452*T^2 - 404801113461840*T - 1483175094148800
$23$
\( T^{9} - 216 T^{8} + \cdots + 22\!\cdots\!44 \)
T^9 - 216*T^8 - 53643*T^7 + 10759163*T^6 + 1089597101*T^5 - 156666082689*T^4 - 11040212833240*T^3 + 558184347978767*T^2 + 37879556564506128*T + 225784040085535744
$29$
\( T^{9} + 91 T^{8} + \cdots + 27\!\cdots\!88 \)
T^9 + 91*T^8 - 174158*T^7 - 17366128*T^6 + 9185210808*T^5 + 899274593664*T^4 - 139084897122816*T^3 - 9110741575389696*T^2 + 564255102670468992*T + 27221149695162175488
$31$
\( T^{9} + 636 T^{8} + \cdots - 56\!\cdots\!52 \)
T^9 + 636*T^8 + 75556*T^7 - 28612128*T^6 - 8281576528*T^5 - 477100859168*T^4 + 57673310843648*T^3 + 7263520911850496*T^2 + 133521957522855936*T - 5656313714465636352
$37$
\( T^{9} + 967 T^{8} + \cdots + 72\!\cdots\!76 \)
T^9 + 967*T^8 + 243580*T^7 - 22130516*T^6 - 12436000368*T^5 + 137596233472*T^4 + 187648286596608*T^3 - 7672354452867072*T^2 - 164994221839429632*T + 7221890531726770176
$41$
\( T^{9} - 226 T^{8} + \cdots + 58\!\cdots\!96 \)
T^9 - 226*T^8 - 282435*T^7 + 63750727*T^6 + 21609891843*T^5 - 6306359254017*T^4 - 202236274629678*T^3 + 191703456737784097*T^2 - 19247276606096634666*T + 583207646675331104496
$43$
\( T^{9} - 42 T^{8} + \cdots + 51\!\cdots\!52 \)
T^9 - 42*T^8 - 302261*T^7 - 10703918*T^6 + 29024570884*T^5 + 2429355295720*T^4 - 945106239520144*T^3 - 110487315152691840*T^2 + 3745159933714018560*T + 514275142180864868352
$47$
\( T^{9} - 269 T^{8} + \cdots + 51\!\cdots\!84 \)
T^9 - 269*T^8 - 606028*T^7 + 72580584*T^6 + 132311976056*T^5 + 2271547150288*T^4 - 10399685852156128*T^3 - 1275770712980367936*T^2 + 51796098595737847296*T + 5180221529050156176384
$53$
\( T^{9} - 1227 T^{8} + \cdots - 12\!\cdots\!32 \)
T^9 - 1227*T^8 - 429690*T^7 + 905955235*T^6 - 3472942078*T^5 - 240660019233381*T^4 + 17941237062097609*T^3 + 28176524689842303366*T^2 - 1656905449020624713028*T - 1265958975712272633130632
$59$
\( T^{9} - 613 T^{8} + \cdots - 15\!\cdots\!52 \)
T^9 - 613*T^8 - 572432*T^7 + 364432436*T^6 + 90481814952*T^5 - 70182997140432*T^4 - 1000333847064864*T^3 + 4507635855927083776*T^2 - 449210477023535234944*T - 15500334992683638845952
$61$
\( T^{9} - 427 T^{8} + \cdots + 49\!\cdots\!64 \)
T^9 - 427*T^8 - 624964*T^7 + 406641104*T^6 - 33043073776*T^5 - 15772751105488*T^4 + 1855682143709632*T^3 + 124644631383491904*T^2 - 19276390395427431552*T + 497471629797562174464
$67$
\( T^{9} - 271 T^{8} + \cdots - 89\!\cdots\!72 \)
T^9 - 271*T^8 - 1689290*T^7 + 444504500*T^6 + 861271633320*T^5 - 258199815903216*T^4 - 133358096368324800*T^3 + 46723867168090464256*T^2 + 1012034209076088406016*T - 897922080360658950684672
$71$
\( T^{9} + 2279 T^{8} + \cdots - 15\!\cdots\!32 \)
T^9 + 2279*T^8 + 13710*T^7 - 3713478448*T^6 - 2812976322272*T^5 + 593150091807408*T^4 + 1436318669482760768*T^3 + 501753468207434631168*T^2 + 8381254664265210043392*T - 15530202419694640792645632
$73$
\( T^{9} + 3602 T^{8} + \cdots + 24\!\cdots\!24 \)
T^9 + 3602*T^8 + 4382917*T^7 + 1215878513*T^6 - 1579613817773*T^5 - 1221939300094063*T^4 - 117610659471505742*T^3 + 118354441151372992799*T^2 + 28715068607783606487238*T + 243994302977203436011224
$79$
\( T^{9} + 1182 T^{8} + \cdots + 19\!\cdots\!00 \)
T^9 + 1182*T^8 - 1819212*T^7 - 1699593072*T^6 + 1303275364544*T^5 + 675537196748256*T^4 - 374192120558897088*T^3 - 76901370480790721152*T^2 + 31579418144400413406720*T + 1939394311564643605811200
$83$
\( T^{9} - 1877 T^{8} + \cdots - 41\!\cdots\!88 \)
T^9 - 1877*T^8 - 2350840*T^7 + 4721478375*T^6 + 1880951108764*T^5 - 3294242954026009*T^4 - 711519704689235001*T^3 + 395593927687194678888*T^2 - 8608879332252716182896*T - 4105367177690789621244288
$89$
\( T^{9} + 1258 T^{8} + \cdots + 94\!\cdots\!48 \)
T^9 + 1258*T^8 - 1799516*T^7 - 2572230784*T^6 + 755047072832*T^5 + 1540586755214240*T^4 + 28581002230567360*T^3 - 259637634627878874240*T^2 - 9561525462383104048128*T + 9411181505703790458267648
$97$
\( T^{9} + 4002 T^{8} + \cdots - 56\!\cdots\!64 \)
T^9 + 4002*T^8 + 3458952*T^7 - 5733497880*T^6 - 11317665267008*T^5 - 3001968418497856*T^4 + 5684446199048870016*T^3 + 4456614018808668869504*T^2 + 840994207715661865478656*T - 56410713245710565816049664
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