
// Make newform 784.3.c.a in Magma, downloaded from the LMFDB on 06 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_784_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_784_3_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_784_3_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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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_784_c();"
function MakeCharacter_784_c()
    N := 784;
    order := 2;
    char_gens := [687, 197, 689];
    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_784_c_Hecke();"
function MakeCharacter_784_c_Hecke(Kf)
    N := 784;
    order := 2;
    char_gens := [687, 197, 689];
    char_values := [[1, 0], [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]];
    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, -1], [0, -1], [0, 0], [-15, 0], [0, 8], [0, 17], [0, 9], [9, 0], [-6, 0], [0, 7], [31, 0], [0, -32], [-10, 0], [0, 25], [-57, 0], [0, 47], [0, 47], [49, 0], [126, 0], [0, -15], [73, 0], [0, 8], [0, -33], [0, 16], [0, 57], [0, 41], [-39, 0], [103, 0], [-78, 0], [-50, 0], [0, 57], [63, 0], [0, -136], [-177, 0], [-55, 0], [0, 57], [-23, 0], [0, 160], [0, -81], [-63, 0], [0, 72], [57, 0], [-209, 0], [-150, 0], [0, 119], [-346, 0], [0, 192], [0, -81], [0, -97], [-273, 0], [222, 0], [0, -127], [0, -56], [0, 111], [153, 0], [0, -263], [0, 41], [359, 0], [-222, 0], [0, -1], [0, 136], [0, -56], [0, -281], [0, -129], [447, 0], [-151, 0], [274, 0], [-543, 0], [0, 104], [0, -175], [-39, 0], [0, -201], [383, 0], [230, 0], [0, 185], [-201, 0], [0, 31], [-177, 0], [0, 193], [0, -344], [-22, 0], [-327, 0], [0, 16], [0, 345], [729, 0], [-270, 0], [799, 0], [0, -248], [814, 0], [0, -231], [0, 311], [857, 0], [-570, 0], [425, 0], [0, -112], [0, 223], [0, -143], [0, 457], [-721, 0], [118, 0], [-633, 0], [0, 159], [447, 0], [-527, 0], [0, 97], [0, -552], [0, 175], [-807, 0], [0, 224], [0, -423], [503, 0], [930, 0], [0, -305], [-194, 0], [-33, 0], [0, 200], [0, -441], [-33, 0], [870, 0], [0, 585], [-14, 0], [0, 159], [273, 0], [0, 89], [906, 0], [703, 0], [0, -343], [0, -16], [0, 559], [577, 0], [-162, 0], [-775, 0], [-950, 0], [0, -113], [0, -528], [0, -103], [0, -193], [0, -504], [-465, 0], [0, -72], [-753, 0], [25, 0], [246, 0], [0, -855], [0, 208], [0, 696], [0, 113], [0, -823], [-855, 0], [-241, 0], [0, -768], [1286, 0], [0, -743], [737, 0], [-1266, 0], [473, 0], [0, -193], [0, 496], [0, -71], [201, 0], [66, 0], [-194, 0], [0, 857], [-1521, 0], [0, -649], [-823, 0], [0, 73]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_784_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_784_3_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_784_3_c_a(:prec:=2)
    chi := MakeCharacter_784_c();
    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_784_3_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_784_3_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![3, 0, 1]>],Snew);
    return Vf;
end function;