
// Make newform 800.4.c.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_800_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_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_800_4_c_l();".
// 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_800_4_c_l();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 4, 0, 2], [0, 8, 0, 2], [12, 0, 8, 0]];
    Rf_basisdens := [1, 1, 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_800_c();"
function MakeCharacter_800_c()
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    v := [2, 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_800_c_Hecke();"
function MakeCharacter_800_c_Hecke(Kf)
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 7, 0], [0, 0, 0, -1], [0, -31, 0, 0], [0, 23, 0, 0], [0, 0, 0, -12], [0, 0, 43, 0], [90, 0, 0, 0], [0, 0, 0, -17], [0, 107, 0, 0], [-10, 0, 0, 0], [0, 0, -15, 0], [0, 0, -89, 0], [0, -339, 0, 0], [0, 0, 0, 46], [250, 0, 0, 0], [0, 0, -11, 0], [0, 0, 0, -41], [0, 261, 0, 0], [0, 0, 0, -98], [0, 0, 85, 0], [-970, 0, 0, 0], [0, 467, 0, 0], [-602, 0, 0, 0], [0, 0, -409, 0], [0, 0, 341, 0], [-2154, 0, 0, 0], [0, -1091, 0, 0], [0, 0, -293, 0], [0, 0, 0, -23], [0, 1047, 0, 0], [0, 0, 0, 154], [334, 0, 0, 0], [0, 0, 0, 351], [0, -417, 0, 0], [0, 0, -691, 0], [0, 0, -1, 0], [0, -919, 0, 0], [0, 0, 0, -206], [1862, 0, 0, 0], [0, 0, 0, -231], [0, 1689, 0, 0], [0, -33, 0, 0], [0, 0, 0, 136], [0, 0, 0, -591], [0, 0, 663, 0], [0, 0, 977, 0], [3250, 0, 0, 0], [0, 1649, 0, 0], [0, 0, 0, 62], [5150, 0, 0, 0], [0, 0, 0, 155], [0, 2083, 0, 0], [0, 0, 215, 0], [1494, 0, 0, 0], [0, 0, 0, 561], [0, 503, 0, 0], [-3210, 0, 0, 0], [0, 0, 813, 0], [0, -1811, 0, 0], [0, 0, -467, 0], [0, 0, 0, -995], [0, 4389, 0, 0], [0, 2523, 0, 0], [0, 0, 0, -35], [0, 1287, 0, 0], [0, 0, -591, 0], [10170, 0, 0, 0], [0, -159, 0, 0], [0, 0, 0, -1384], [0, 0, 687, 0], [0, -1639, 0, 0], [0, 0, 0, -572], [0, 0, -257, 0], [-834, 0, 0, 0], [0, 4367, 0, 0], [242, 0, 0, 0], [6514, 0, 0, 0], [0, 0, 0, -1798], [7250, 0, 0, 0], [0, 0, 0, 557], [0, 5741, 0, 0], [0, 0, 0, -424], [0, 0, -167, 0], [1306, 0, 0, 0], [0, 4763, 0, 0], [1518, 0, 0, 0], [0, 0, 3867, 0], [0, 0, 3797, 0], [0, 0, 0, -426], [0, 0, -293, 0], [0, 0, 0, 331], [0, 0, 0, 2160], [0, 0, 687, 0], [-18550, 0, 0, 0], [-2102, 0, 0, 0], [0, 0, 3957, 0], [-9922, 0, 0, 0], [0, 0, -831, 0], [0, 7547, 0, 0], [0, 0, 1265, 0], [5906, 0, 0, 0], [0, 0, 0, -547], [0, 6643, 0, 0], [0, 0, -2019, 0], [0, 5721, 0, 0], [0, 0, 0, 1582], [3110, 0, 0, 0], [0, 0, 2503, 0], [0, -2671, 0, 0], [0, -9857, 0, 0], [0, 0, 0, 1472], [0, 0, 0, 1371], [-2690, 0, 0, 0], [0, 0, 2737, 0], [0, 0, 4019, 0], [0, -1739, 0, 0], [0, 0, 0, 1182], [-110, 0, 0, 0], [0, -7139, 0, 0], [0, -9193, 0, 0], [0, 0, 3425, 0], [0, 0, 0, 1063], [-15830, 0, 0, 0], [20050, 0, 0, 0], [0, 0, 0, -2362], [0, 0, 2535, 0], [0, -8599, 0, 0], [0, 0, 0, -514], [0, 0, -541, 0], [0, 0, 0, 831], [0, -5737, 0, 0], [31802, 0, 0, 0], [5310, 0, 0, 0], [0, 18969, 0, 0], [0, 0, -8415, 0], [0, 8763, 0, 0], [-8970, 0, 0, 0], [0, 0, 0, -397], [-15550, 0, 0, 0], [0, 0, -5973, 0], [0, 0, 193, 0], [-19066, 0, 0, 0], [0, 0, 0, 5338], [0, -7231, 0, 0], [0, -14673, 0, 0], [0, 0, 0, -2550], [0, 0, 5535, 0], [0, 16063, 0, 0], [-33570, 0, 0, 0], [0, 0, -1439, 0], [0, 0, 10471, 0], [0, 0, -2679, 0], [0, 0, 0, -2705], [0, 0, 0, 4194], [21994, 0, 0, 0], [0, 8143, 0, 0], [24302, 0, 0, 0], [0, 0, -4443, 0], [0, -11211, 0, 0], [0, 0, -9789, 0], [0, 0, 0, -2875], [0, -14493, 0, 0], [0, 0, 7183, 0], [0, 0, 0, -1341], [0, 203, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_800_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_800_4_c_l();".
// 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_800_4_c_l(:prec:=4)
    chi := MakeCharacter_800_c();
    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_800_4_c_l();".
// 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_800_4_c_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_800_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![20, 0, 1]>,<11,R![-80, 0, 1]>],Snew);
    return Vf;
end function;