
// Make newform 224.3.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_d_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_224_3_d_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_224_3_d_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 := [4, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -1, 0, 1], [0, 5, 0, -1], [-3, 0, 2, 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_224_d();"
function MakeCharacter_224_d()
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    v := [1, 2, 2];
    // 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_224_d_Hecke();"
function MakeCharacter_224_d_Hecke(Kf)
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 1, 0, 0], [-2, 0, 1, 0], [0, 0, 0, -1], [0, -2, 0, 4], [-2, 0, 3, 0], [-14, 0, 2, 0], [0, -5, 0, 8], [0, 6, 0, 4], [10, 0, 6, 0], [0, -2, 0, 0], [34, 0, 2, 0], [-30, 0, -2, 0], [0, 6, 0, -20], [0, -18, 0, -8], [-70, 0, 4, 0], [0, -9, 0, -16], [-2, 0, -3, 0], [0, 12, 0, -20], [0, -32, 0, 16], [-38, 0, -16, 0], [0, 0, 0, -40], [0, 31, 0, 24], [34, 0, 8, 0], [-150, 0, -2, 0], [22, 0, -11, 0], [0, -62, 0, -8], [0, -44, 0, -4], [-62, 0, -10, 0], [14, 0, 24, 0], [0, -90, 0, -12], [0, 67, 0, 40], [2, 0, 44, 0], [0, 65, 0, -24], [50, 0, -4, 0], [0, 94, 0, 28], [-34, 0, 3, 0], [0, -134, 0, -4], [0, 18, 0, 80], [110, 0, -33, 0], [0, -8, 0, 92], [-234, 0, 3, 0], [0, -68, 0, 48], [6, 0, -56, 0], [-54, 0, 16, 0], [0, 30, 0, -8], [0, -36, 0, 76], [0, -24, 0, -104], [0, -29, 0, -8], [190, 0, -5, 0], [18, 0, -16, 0], [0, -18, 0, -116], [10, 0, -26, 0], [0, 113, 0, -56], [2, 0, -24, 0], [0, 40, 0, -32], [-194, 0, 23, 0], [0, -148, 0, -8], [-14, 0, 40, 0], [-78, 0, 40, 0], [0, -57, 0, -112], [-42, 0, -55, 0], [0, -85, 0, 88], [0, -24, 0, 40], [-142, 0, -42, 0], [90, 0, 16, 0], [0, 126, 0, 92], [198, 0, -40, 0], [0, -146, 0, -36], [278, 0, 51, 0], [-70, 0, 20, 0], [0, 274, 0, 20], [0, -56, 0, 184], [18, 0, 28, 0], [0, 98, 0, 44], [0, 70, 0, -104], [178, 0, 34, 0], [214, 0, 59, 0], [14, 0, 116, 0], [266, 0, -18, 0], [0, -203, 0, -48], [-62, 0, 24, 0], [0, 26, 0, 52], [2, 0, 22, 0], [0, -316, 0, 48], [0, 108, 0, -92], [146, 0, 40, 0], [-474, 0, 16, 0], [550, 0, -57, 0], [0, 24, 0, 64], [0, 239, 0, 136], [0, 94, 0, -32], [0, -10, 0, 52], [0, 276, 0, -76], [0, -8, 0, 148], [0, -208, 0, -208], [-314, 0, -117, 0], [298, 0, 114, 0], [0, -61, 0, 96], [-638, 0, -12, 0], [0, -234, 0, -148], [314, 0, -56, 0], [0, -87, 0, -80], [-738, 0, 44, 0], [0, -6, 0, 172], [-614, 0, 92, 0], [0, -27, 0, -8], [-46, 0, -68, 0], [0, -144, 0, -40], [274, 0, 144, 0], [0, -8, 0, 304], [-38, 0, -82, 0], [-458, 0, -72, 0], [0, 319, 0, 64], [0, 348, 0, -120], [142, 0, 124, 0], [0, -87, 0, -272], [0, -74, 0, 8], [-726, 0, 2, 0], [0, 6, 0, 332], [102, 0, 65, 0], [642, 0, -44, 0], [-234, 0, 131, 0], [0, 136, 0, 44], [0, -333, 0, 184], [-726, 0, -6, 0], [-46, 0, 162, 0], [0, 250, 0, 152], [0, -10, 0, 192], [-98, 0, 117, 0], [0, 410, 0, -132], [0, -198, 0, 20], [0, -86, 0, 52], [-182, 0, 238, 0], [-478, 0, 74, 0], [-918, 0, -70, 0], [-50, 0, 29, 0], [0, 191, 0, 312], [750, 0, 79, 0], [742, 0, -88, 0], [0, -141, 0, 240], [1042, 0, 48, 0], [0, 156, 0, -176], [0, -228, 0, 60], [-178, 0, 13, 0], [0, 330, 0, 88], [-834, 0, -29, 0], [-598, 0, -82, 0], [0, -655, 0, -88], [0, -20, 0, -160], [-478, 0, 98, 0], [-1254, 0, 24, 0], [0, -180, 0, -108], [0, 502, 0, -240], [0, 104, 0, 212], [0, 146, 0, 156], [0, -268, 0, -32], [-590, 0, 90, 0], [1050, 0, 0, 0], [-58, 0, -231, 0], [0, 270, 0, 140], [546, 0, -144, 0], [0, -770, 0, -12], [0, -377, 0, -288], [-318, 0, -244, 0], [0, 522, 0, 296], [0, 292, 0, -40], [-1586, 0, -11, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_d_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_224_3_d_a(:prec:=4)
    chi := MakeCharacter_224_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_224_3_d_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_224_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![4, 0, 1]>],Snew);
    return Vf;
end function;