[N,k,chi] = [378,4,Mod(109,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.109");
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 \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\)
\(29\)
\(325\)
\(\chi(n)\)
\(1\)
\(\beta_{1}\)
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
\( T_{5}^{8} + 4T_{5}^{7} + 310T_{5}^{6} + 48T_{5}^{5} + 85860T_{5}^{4} + 155736T_{5}^{3} + 1263600T_{5}^{2} - 1850688T_{5} + 9144576 \)
T5^8 + 4*T5^7 + 310*T5^6 + 48*T5^5 + 85860*T5^4 + 155736*T5^3 + 1263600*T5^2 - 1850688*T5 + 9144576
acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 4)^{4} \)
(T^2 - 2*T + 4)^4
$3$
\( T^{8} \)
T^8
$5$
\( T^{8} + 4 T^{7} + 310 T^{6} + \cdots + 9144576 \)
T^8 + 4*T^7 + 310*T^6 + 48*T^5 + 85860*T^4 + 155736*T^3 + 1263600*T^2 - 1850688*T + 9144576
$7$
\( T^{8} - 25 T^{7} + \cdots + 13841287201 \)
T^8 - 25*T^7 + 892*T^6 - 12593*T^5 + 338198*T^4 - 4319399*T^3 + 104942908*T^2 - 1008840175*T + 13841287201
$11$
\( T^{8} + 56 T^{7} + \cdots + 943105130496 \)
T^8 + 56*T^7 + 4030*T^6 + 121224*T^5 + 6566436*T^4 + 185332968*T^3 + 6466699152*T^2 + 83171971584*T + 943105130496
$13$
\( (T^{4} + 9 T^{3} - 2637 T^{2} + \cdots + 1313022)^{2} \)
(T^4 + 9*T^3 - 2637*T^2 - 1499*T + 1313022)^2
$17$
\( T^{8} - 118 T^{7} + \cdots + 8822183086656 \)
T^8 - 118*T^7 + 15394*T^6 - 317292*T^5 + 34085484*T^4 - 1061673696*T^3 + 55843163856*T^2 - 728819721216*T + 8822183086656
$19$
\( T^{8} - 37 T^{7} + \cdots + 96\!\cdots\!84 \)
T^8 - 37*T^7 + 23269*T^6 - 73852*T^5 + 397935340*T^4 - 2427135472*T^3 + 2342320426576*T^2 + 43337361015872*T + 9610169502486784
$23$
\( T^{8} + 200 T^{7} + \cdots + 10\!\cdots\!44 \)
T^8 + 200*T^7 + 67936*T^6 + 9369792*T^5 + 2597811984*T^4 + 337596339456*T^3 + 46941095490048*T^2 + 2405777480437248*T + 103486185512665344
$29$
\( (T^{4} - 262 T^{3} - 27174 T^{2} + \cdots - 737431128)^{2} \)
(T^4 - 262*T^3 - 27174*T^2 + 11435400*T - 737431128)^2
$31$
\( T^{8} - 276 T^{7} + \cdots + 11\!\cdots\!69 \)
T^8 - 276*T^7 + 144984*T^6 - 27301720*T^5 + 12208327665*T^4 - 2191790480136*T^3 + 461070628112200*T^2 - 25122839822284668*T + 1178072183652973569
$37$
\( T^{8} + 185 T^{7} + \cdots + 26\!\cdots\!16 \)
T^8 + 185*T^7 + 174430*T^6 - 513151*T^5 + 16814713366*T^4 - 139632274645*T^3 + 889902495449839*T^2 - 66034751748186298*T + 26983040669216224516
$41$
\( (T^{4} + 30 T^{3} - 329454 T^{2} + \cdots + 23111436456)^{2} \)
(T^4 + 30*T^3 - 329454*T^2 + 4343976*T + 23111436456)^2
$43$
\( (T^{4} + 778 T^{3} + 137328 T^{2} + \cdots - 309283361)^{2} \)
(T^4 + 778*T^3 + 137328*T^2 - 5198522*T - 309283361)^2
$47$
\( T^{8} + 30 T^{7} + \cdots + 47\!\cdots\!36 \)
T^8 + 30*T^7 + 181314*T^6 - 813348*T^5 + 25749496620*T^4 + 2746436544*T^3 + 1244497662049680*T^2 - 15794824891602816*T + 47179051449215132736
$53$
\( T^{8} + 480 T^{7} + \cdots + 21\!\cdots\!24 \)
T^8 + 480*T^7 + 462456*T^6 + 64374912*T^5 + 91427199984*T^4 + 15971896659456*T^3 + 8791812994151808*T^2 - 404745768981199872*T + 21211710087695341824
$59$
\( T^{8} - 296 T^{7} + \cdots + 17\!\cdots\!64 \)
T^8 - 296*T^7 + 596422*T^6 - 147655944*T^5 + 307213944804*T^4 - 78357907230696*T^3 + 20109484073279952*T^2 - 624494042307262080*T + 17535543244135179264
$61$
\( T^{8} - 474 T^{7} + \cdots + 37\!\cdots\!61 \)
T^8 - 474*T^7 + 829710*T^6 - 516069208*T^5 + 617588161275*T^4 - 300937936348296*T^3 + 124088673938047990*T^2 - 24585560931782452122*T + 3750987792559258005561
$67$
\( T^{8} - 1319 T^{7} + \cdots + 35\!\cdots\!64 \)
T^8 - 1319*T^7 + 1638946*T^6 - 532865791*T^5 + 254941997090*T^4 + 70146682242941*T^3 + 38067754843873429*T^2 + 3788896144157377876*T + 359089375046729230864
$71$
\( (T^{4} - 926 T^{3} + \cdots - 55652552064)^{2} \)
(T^4 - 926*T^3 - 215778*T^2 + 342601848*T - 55652552064)^2
$73$
\( T^{8} + 1423 T^{7} + \cdots + 42\!\cdots\!44 \)
T^8 + 1423*T^7 + 1707247*T^6 + 750680366*T^5 + 333945909956*T^4 + 11071514166272*T^3 + 28823809975389184*T^2 + 3069312905673441280*T + 422578889771503058944
$79$
\( T^{8} - 765 T^{7} + \cdots + 25\!\cdots\!36 \)
T^8 - 765*T^7 + 1231632*T^6 - 466450147*T^5 + 786996568608*T^4 - 313016521225977*T^3 + 229828766220146293*T^2 - 764249267153592444*T + 2530038403341989136
$83$
\( (T^{4} - 830 T^{3} - 10644 T^{2} + \cdots + 331327584)^{2} \)
(T^4 - 830*T^3 - 10644*T^2 + 8136360*T + 331327584)^2
$89$
\( T^{8} - 864 T^{7} + \cdots + 10\!\cdots\!04 \)
T^8 - 864*T^7 + 816750*T^6 - 303170040*T^5 + 194878445940*T^4 - 69375715920648*T^3 + 30799349784598896*T^2 - 5958581164416933696*T + 1072641045893965519104
$97$
\( (T^{4} - 544 T^{3} + \cdots - 91533471583)^{2} \)
(T^4 - 544*T^3 - 973698*T^2 + 632425216*T - 91533471583)^2
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