
// Make newform 675.2.ba.c in Magma, downloaded from the LMFDB on 04 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_ba();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_675_2_ba_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_675_ba();"
function MakeCharacter_675_ba()
    N := 675;
    order := 36;
    char_gens := [326, 352];
    v := [10, 9];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_675_ba_Hecke( : Kf := Kf)
    return MakeCharacter_675_ba();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_675_2_ba_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_675_2_ba_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_ba();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![17583975504331247446360641, 0, 0, 0, 305502953339615277379932500247, 0, 0, 0, 2275201857570289234344248514858009, 0, 0, 0, 434505042143136066650865195362072796, 0, 0, 0, 1017459100853979225223914037858802121000, 0, 0, 0, -117478103981662609048953105923519911763847, 0, 0, 0, 3446491174021073052703320040439244477148257, 0, 0, 0, -12700787004694682735112896464728009568339776, 0, 0, 0, 3116999190199952560502905529014433122499504439, 0, 0, 0, 17410293181578210781982928396800954941982053647, 0, 0, 0, -35175383141410752566336003266312961548051964334, 0, 0, 0, -6985709748049829693998197288705390000917127719685, 0, 0, 0, 15966082115394789368091568805547960005554044767031, 0, 0, 0, 35113025188392494603105551195732975272901150821171, 0, 0, 0, 5019891216070280554285340574629636396251640110898010, 0, 0, 0, -2702749807275468174060745175816590390216509757637460, 0, 0, 0, 32024988362916287109853139016784622555507334335917806, 0, 0, 0, -1928257511205493214644737366422638515880947677265513561, 0, 0, 0, 10126342470763277311239868687809089297450812735232488122, 0, 0, 0, -13732160989036719952105955333417654349358222744620870789, 0, 0, 0, -9339312318647955132522838769484419346812493813050546484, 0, 0, 0, 2872549517900912880479113729491620041198678750483804566, 0, 0, 0, 68116604382084281408309547611073354006450337510149190407, 0, 0, 0, -33861429739436800739108672388041258810969537018579336832, 0, 0, 0, 3358133915831060583122704075463707381516589102742499020, 0, 0, 0, -29091725299201722545136303486960471875079842070149469351, 0, 0, 0, 4585327358018488039855809596901737563227246738321204813, 0, 0, 0, 9367043772234954426260566037004043496002098045700412445, 0, 0, 0, 8816438330244189737855077069707251857604782280724241566, 0, 0, 0, 2582758382427450528062959571446072760230093655583619812, 0, 0, 0, 817374165635976854504107114221197176273284082117080990, 0, 0, 0, 50255187638927768218411435128958512943514758250246523, 0, 0, 0, -410356842676969762518595111453784829249142886888223, 0, 0, 0, -5601187379870577949456089195876132838455512137142376, 0, 0, 0, -442752742332357875363004471009389924255637333967869, 0, 0, 0, -29812519976467226824035273157839572258760988355091, 0, 0, 0, 31289807108078707837049521660192869734704501599765, 0, 0, 0, 1555454498548116833430736224632625633731423512743, 0, 0, 0, 208478713983427702463966887225326533521149472587, 0, 0, 0, -80300072383775347485666409048062579856196992446, 0, 0, 0, -1558057893110029336380728722159181375902997700, 0, 0, 0, -589473969738210250712354955394780279650702531, 0, 0, 0, 166837888129275458598775170534142012793333181, 0, 0, 0, -1892998757261187508159448870099480582849739, 0, 0, 0, 547014386337515072449298509042317347579340, 0, 0, 0, -86187847224465580400740090681102081532466, 0, 0, 0, -1476020685216054232389609869537002777290, 0, 0, 0, -25354608941990843801080616367775167588, 0, 0, 0, 24975529931301040625721480677716505145, 0, 0, 0, 650323347997569056099107499744648631, 0, 0, 0, 15555978252984297924578975299871373, 0, 0, 0, -2789980253350298221443257100827016, 0, 0, 0, -75327788390879147597985448278945, 0, 0, 0, -1044933190672447735697640703554, 0, 0, 0, 221893566663959998473700080657, 0, 0, 0, 4174673285720857414489983288, 0, 0, 0, 44719248245060953869378318, 0, 0, 0, -9674858305595559593268756, 0, 0, 0, -125055573384729857050413, 0, 0, 0, -981209696795291524068, 0, 0, 0, 304757316165521131479, 0, 0, 0, 1528774138214704125, 0, 0, 0, 2653304474384208, 0, 0, 0, -5157229056700500, 0, 0, 0, 8603348372073, 0, 0, 0, -371411828577, 0, 0, 0, 60450329646, 0, 0, 0, -160478280, 0, 0, 0, 200934, 0, 0, 0, -286956, 0, 0, 0, 468, 0, 0, 0, 0, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;