
// Make newform 50.8.a.e in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_50_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_50_8_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_50_8_a_e();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_50_a();"
function MakeCharacter_50_a()
    N := 50;
    order := 1;
    char_gens := [27];
    v := [1];
    // 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_50_a_Hecke( : Kf := Kf)
    return MakeCharacter_50_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 8;
    raw_aps := [[8], [-87], [0], [-1366], [-1083], [5468], [25269], [33485], [5838], [125280], [-73798], [-395926], [-22683], [100148], [1145244], [-354882], [1098360], [-422998], [2558579], [-2287428], [6372443], [-2019250], [7972983], [2185935], [-5823646], [433062], [-10429492], [21316389], [16030430], [905463], [16035194], [3830772], [9693429], [44940605], [-49420800], [20107802], [78669434], [-69974917], [-83891796], [-27366792], [-79763415], [52871402], [-103225338], [-49965067], [-33385626], [92510660], [56115347], [-77993572], [131342484], [228442340], [377948598], [-113186160], [-200235943], [-462823173], [-487751346], [-401821122], [455386320], [508458602], [646826864], [189546462], [463383803], [123173358], [564202169], [551416662], [-171415702], [-1033492596], [24123257], [-875079661], [-83244891], [397606670], [-293709882], [-171667050], [-955463356], [344673458], [1452725255], [-1878392802], [-1126315770], [-1457334316], [1958492517], [-1255936675], [-44654775], [-58312108], [2158067862], [2028669173], [-2516405080], [-2567198457], [828488505], [345538169], [481628172], [1183069868], [1011914004], [3786731670], [-720259126], [-1796669628], [2545397660], [-1314007452], [-2503855830], [-1220282283], [1933184273], [-2549203888], [4582911689], [-4942073076], [782258448], [1196276025], [2410706492], [-6453941341], [527193759], [-8990563257], [3407202390], [-4008079453], [3611514704], [416268038], [-232065666], [-1509137620], [11851212602], [-3894752118], [-687186652], [-2241571956], [3445687518], [7597527855], [-549900688], [72947258], [-3970034256], [-11894291247], [2089482857], [12181797372], [3807262820], [2744255130], [515574464], [9167913788], [-7951336900], [12033305358], [-14665047088], [-18675696496], [14550158517], [7455064685], [18567068808], [-217239796], [10402077714], [15401977170], [9379140092], [-14918954718], [17290505228], [-192632721], [15519895940], [10410809040], [-15450604462], [2503113189], [20584244525], [19107304248], [33306337124], [-9313125138], [4385142833], [-17876414436], [-2847419236], [-38464295028], [10737538190], [5344928670], [-14812454461], [21873447372], [24516334104], [13405631073], [-4647400816], [-32485958283], [-6065058111], [-21084216402], [42647626382], [-15922634716]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_50_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_50_8_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_50_8_a_e(:prec:=1)
    chi := MakeCharacter_50_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_50_8_a_e();".
// 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_50_8_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_50_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<3,R![87, 1]>],Snew);
    return Vf;
end function;