
// Make newform 100.4.a.c in Magma, downloaded from the LMFDB on 14 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_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_100_4_a_c();".
// 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_100_4_a_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 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_100_a();"
function MakeCharacter_100_a()
    N := 100;
    order := 1;
    char_gens := [51, 77];
    v := [1, 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_100_a_Hecke( : Kf := Kf)
    return MakeCharacter_100_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 := 4;
    raw_aps := [[0], [1], [0], [26], [45], [44], [117], [-91], [-18], [144], [26], [-214], [-459], [-460], [-468], [558], [-72], [-118], [251], [108], [299], [-898], [927], [351], [386], [-954], [-772], [1197], [-802], [1143], [-2374], [-1260], [-891], [389], [1296], [-2710], [-1846], [1475], [-1476], [1368], [-1503], [3770], [-4122], [-1963], [2934], [1412], [3419], [-100], [4212], [-3484], [918], [3744], [-4231], [-2925], [-18], [-6786], [7632], [650], [-3232], [4446], [2483], [-4050], [2321], [-3258], [3626], [3852], [7553], [-109], [-2835], [2990], [9126], [-9594], [9764], [6722], [-13537], [-8658], [-8874], [5876], [-1755], [4589], [5409], [12116], [9126], [629], [4472], [-3393], [-5031], [-6487], [2700], [-2932], [-15660], [10134], [1898], [-6300], [18044], [-6876], [-4806], [7749], [8153], [-10576], [7553], [13500], [23184], [-8055], [3068], [12419], [-12393], [23751], [-11610], [26675], [17264], [-26026], [5022], [7820], [15002], [-918], [-23452], [20556], [30654], [8919], [-22912], [-1222], [144], [-12519], [11873], [-3060], [4004], [10314], [9872], [7436], [-16900], [-23058], [-8224], [-7696], [-6363], [8333], [32760], [-43732], [16866], [16146], [32444], [-2574], [-27604], [11655], [33428], [-17712], [-10270], [38709], [15509], [15912], [-10972], [-18738], [-21367], [20124], [23132], [31212], [-6994], [19422], [11699], [-42948], [-3816], [-43407], [-43216], [-47619], [-4671], [9054], [8126], [38468]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_4_a_c();".
// 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_100_4_a_c(:prec:=1)
    chi := MakeCharacter_100_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_100_4_a_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_100_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-1, 1]>],Snew);
    return Vf;
end function;