
// Make newform 100.6.c.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_100_c_Hecke();"
// 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_6_c_a();".
// 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_6_c_a();".
// 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
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_100_c();"
function MakeCharacter_100_c()
    N := 100;
    order := 2;
    char_gens := [51, 77];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_100_c_Hecke();"
function MakeCharacter_100_c_Hecke(Kf)
    N := 100;
    order := 2;
    char_gens := [51, 77];
    char_values := [[1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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+1)]];
    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 := 6;
    raw_aps := [[0, 0], [0, 11], [0, 0], [0, -109], [-480, 0], [0, -311], [0, -93], [1204, 0], [0, -1593], [-5526, 0], [9356, 0], [0, -2809], [-14394, 0], [0, -185], [0, -8073], [0, -2187], [11748, 0], [13202, 0], [0, 5771], [-29532, 0], [0, 16849], [-31208, 0], [0, -19233], [-119514, 0], [0, -47329], [101046, 0], [0, -71717], [0, 28527], [3118, 0], [0, -27267], [0, -12349], [236640, 0], [0, 11079], [193204, 0], [-448554, 0], [-140860, 0], [0, 167939], [0, -50825], [0, -69621], [0, -132507], [142812, 0], [109670, 0], [294948, 0], [0, 501517], [0, 411999], [906712, 0], [506384, 0], [0, -271025], [0, 724287], [478786, 0], [0, 187053], [-169416, 0], [-353746, 0], [1255200, 0], [0, 564387], [0, -131541], [1187742, 0], [431300, 0], [0, -371557], [1921926, 0], [0, -815357], [0, 35625], [0, 808811], [-682788, 0], [0, -1352219], [0, -1301733], [-661432, 0], [0, -858529], [0, -65685], [-3509510, 0], [0, 1109961], [-4397304, 0], [0, 1149119], [0, -867803], [5391148, 0], [0, 1636407], [-603114, 0], [0, 374711], [5313570, 0], [-999326, 0], [-2037396, 0], [-5114614, 0], [-3304044, 0], [0, -1008191], [-6583208, 0], [0, -2408913], [6203994, 0], [0, -1421917], [-1756050, 0], [0, 3834283], [0, 679515], [2027064, 0], [0, -1232137], [10384800, 0], [-6494156, 0], [0, -5178261], [-5873046, 0], [2172954, 0], [0, 536803], [7090334, 0], [0, -3348817], [0, -5950425], [0, 4378179], [1156770, 0], [-7078072, 0], [0, 1567019], [0, -5691633], [0, -7932759], [15099840, 0], [-8087050, 0], [0, 355199], [0, 2982169], [0, -7421613], [18204220, 0], [1094612, 0], [7440462, 0], [0, -5395757], [0, 4814991], [0, -5000931], [4010604, 0], [12033818, 0], [0, 1006153], [0, -8120481], [0, 2314551], [11679368, 0], [19954290, 0], [4883314, 0], [13577784, 0], [0, -3212053], [0, 4542781], [-20245700, 0], [0, -2724153], [-1140724, 0], [0, -9507661], [22355082, 0], [10070398, 0], [0, -2029815], [0, 8612783], [0, 10539651], [18787686, 0], [-13245616, 0], [-7669254, 0], [0, -4413929], [0, 15346155], [-32841362, 0], [8421168, 0], [0, 11756305], [0, 5652519], [-10007756, 0], [0, 13071447], [0, 960203], [-25659378, 0], [0, -10282157], [0, -15843381], [0, -1984813], [-13794468, 0], [-8081264, 0], [-29995638, 0], [0, 10381019], [-3476418, 0], [0, -933501], [0, -19275027], [0, -8243761], [-23697624, 0], [0, 28879479], [0, -5505141], [37080716, 0], [0, -2264677]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_6_c_a();".
// 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_6_c_a(:prec:=2)
    chi := MakeCharacter_100_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_6_c_a();".
// 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_6_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![484, 0, 1]>],Snew);
    return Vf;
end function;