[N,k,chi] = [19,8,Mod(1,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.1");
S:= CuspForms(chi, 8);
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
\(19\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 15T_{2}^{5} - 450T_{2}^{4} + 4650T_{2}^{3} + 64272T_{2}^{2} - 289800T_{2} - 1974784 \)
T2^6 - 15*T2^5 - 450*T2^4 + 4650*T2^3 + 64272*T2^2 - 289800*T2 - 1974784
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\).
$p$
$F_p(T)$
$2$
\( T^{6} - 15 T^{5} - 450 T^{4} + \cdots - 1974784 \)
T^6 - 15*T^5 - 450*T^4 + 4650*T^3 + 64272*T^2 - 289800*T - 1974784
$3$
\( T^{6} - 40 T^{5} + \cdots - 15379814304 \)
T^6 - 40*T^5 - 8719*T^4 + 245290*T^3 + 22871940*T^2 - 322311240*T - 15379814304
$5$
\( T^{6} - 219 T^{5} + \cdots - 18322855416000 \)
T^6 - 219*T^5 - 183611*T^4 + 37247463*T^3 + 4116297310*T^2 - 212819831400*T - 18322855416000
$7$
\( T^{6} - 2105 T^{5} + \cdots + 12\!\cdots\!64 \)
T^6 - 2105*T^5 + 14314*T^4 + 2426151150*T^3 - 1618533285531*T^2 + 285563386083915*T + 1241529811449264
$11$
\( T^{6} - 7257 T^{5} + \cdots - 86\!\cdots\!56 \)
T^6 - 7257*T^5 - 58576981*T^4 + 368847603441*T^3 + 1114515371338152*T^2 - 3996248090894353044*T - 8630238040718909522256
$13$
\( T^{6} - 6850 T^{5} + \cdots - 69\!\cdots\!44 \)
T^6 - 6850*T^5 - 284994377*T^4 + 1444968886290*T^3 + 24854386900181016*T^2 - 69177524830693794720*T - 690652649812597748578944
$17$
\( T^{6} - 5415 T^{5} + \cdots + 17\!\cdots\!74 \)
T^6 - 5415*T^5 - 998120776*T^4 + 9513309682830*T^3 - 7068543469513587*T^2 - 114013391989696169895*T + 170160030426523581358074
$19$
\( (T + 6859)^{6} \)
(T + 6859)^6
$23$
\( T^{6} + 720 T^{5} + \cdots - 52\!\cdots\!96 \)
T^6 + 720*T^5 - 14218296849*T^4 - 79930062449640*T^3 + 52258446517371309888*T^2 + 149173131729458706992640*T - 52196894086229652217757802496
$29$
\( T^{6} - 381624 T^{5} + \cdots + 28\!\cdots\!96 \)
T^6 - 381624*T^5 + 17099079015*T^4 + 7360423866796320*T^3 - 600648549687755360760*T^2 - 33971405757528600101492544*T + 2820690004681374239204466078896
$31$
\( T^{6} - 264080 T^{5} + \cdots + 18\!\cdots\!84 \)
T^6 - 264080*T^5 - 87862602432*T^4 + 24420309823899136*T^3 + 1458341173208191618816*T^2 - 540097424017099156209156096*T + 18831418878808873791604742160384
$37$
\( T^{6} - 1082300 T^{5} + \cdots + 47\!\cdots\!16 \)
T^6 - 1082300*T^5 + 144427963596*T^4 + 168583699857480160*T^3 - 51031310383103635400336*T^2 + 1657515556166388608550634560*T + 47545252879122401726993650871616
$41$
\( T^{6} - 485232 T^{5} + \cdots - 50\!\cdots\!24 \)
T^6 - 485232*T^5 - 477475608872*T^4 + 169022418349336800*T^3 + 16072453268710619470864*T^2 + 95116872135737540390431872*T - 5038012649705086333496147226624
$43$
\( T^{6} - 198705 T^{5} + \cdots - 10\!\cdots\!96 \)
T^6 - 198705*T^5 - 796369891581*T^4 + 152390939240532605*T^3 + 112607340202360736435544*T^2 + 2275430363041637301840945360*T - 1030456259867689788159393143876096
$47$
\( T^{6} + 247125 T^{5} + \cdots - 22\!\cdots\!56 \)
T^6 + 247125*T^5 - 1134282980133*T^4 - 84277929348135225*T^3 + 317625887559244962883464*T^2 + 5127789979838386022241916800*T - 22326962577728683069127332555001856
$53$
\( T^{6} - 3226770 T^{5} + \cdots + 63\!\cdots\!64 \)
T^6 - 3226770*T^5 + 6680580671*T^4 + 6839918476502677530*T^3 - 2798062132165543364002344*T^2 - 2944876886827148907627861667680*T + 636208232230193799488402557867227264
$59$
\( T^{6} - 2305380 T^{5} + \cdots - 16\!\cdots\!76 \)
T^6 - 2305380*T^5 - 3791739419959*T^4 + 6049042205847651774*T^3 + 6092914166844871603656804*T^2 - 33997759089326621244804324888*T - 165710809413450950084673012482976
$61$
\( T^{6} - 585731 T^{5} + \cdots - 49\!\cdots\!84 \)
T^6 - 585731*T^5 - 12983898791299*T^4 + 978501154553663707*T^3 + 37870077061975848726590342*T^2 + 11426789022197798787811823912068*T - 4949994176054133392438821192246562984
$67$
\( T^{6} + 3264030 T^{5} + \cdots - 48\!\cdots\!04 \)
T^6 + 3264030*T^5 - 17958621512055*T^4 - 50264895340223308360*T^3 + 80601193193766723963002016*T^2 + 184324908315816432224590122499200*T - 48971289076702013434202564543296834304
$71$
\( T^{6} - 6833682 T^{5} + \cdots + 63\!\cdots\!24 \)
T^6 - 6833682*T^5 + 9213927385656*T^4 + 12658180453581113136*T^3 - 19847080051339622745625776*T^2 - 8314341992455818588229063965600*T + 6352966656568664263248617603655085824
$73$
\( T^{6} + 4160625 T^{5} + \cdots + 42\!\cdots\!74 \)
T^6 + 4160625*T^5 - 18976001363124*T^4 - 76609185100069199150*T^3 - 55579924661475309306986679*T^2 + 10904960971208911754323479354765*T + 420376232319909317261066396138066074
$79$
\( T^{6} + 8680576 T^{5} + \cdots - 74\!\cdots\!96 \)
T^6 + 8680576*T^5 - 10779053820692*T^4 - 311729499220233658464*T^3 - 1103855551176610759232145024*T^2 - 1522481170998457882209201101263872*T - 743096392300159732559219500684743376896
$83$
\( T^{6} + 3785040 T^{5} + \cdots + 37\!\cdots\!16 \)
T^6 + 3785040*T^5 - 59657687927732*T^4 - 92528565619411420320*T^3 + 912997708094146989684145408*T^2 - 1224092160937941926014257811023360*T + 370698187579442177457809872088287958016
$89$
\( T^{6} - 12473466 T^{5} + \cdots + 65\!\cdots\!96 \)
T^6 - 12473466*T^5 + 14002349517856*T^4 + 184171254348156004992*T^3 - 43067397934486103777431808*T^2 - 224207154292159756201724985165312*T + 6516011262179829266519037744891396096
$97$
\( T^{6} - 882830 T^{5} + \cdots - 44\!\cdots\!44 \)
T^6 - 882830*T^5 - 275851158253972*T^4 - 327382740217212672200*T^3 + 19931596828093873713438261056*T^2 + 20578151862483689046935616602283520*T - 440643829592225610917320428354144578514944
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