
// Make newform 784.4.f.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_784_f_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_4_f_f();".
// 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_4_f_f();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [7, 0, 2, 0], [0, 14, 0, 1]];
    Rf_basisdens := [1, 7, 7, 7];
    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_f();"
function MakeCharacter_784_f()
    N := 784;
    order := 2;
    char_gens := [687, 197, 689];
    v := [1, 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_f_Hecke();"
function MakeCharacter_784_f_Hecke(Kf)
    N := 784;
    order := 2;
    char_gens := [687, 197, 689];
    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, 1, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 0, 0, -15], [0, 0, -30, 0], [0, 0, 5, 0], [0, 25, 0, 0], [0, 0, 0, 11], [-168, 0, 0, 0], [0, -85, 0, 0], [-245, 0, 0, 0], [0, 0, -26, 0], [0, 0, 0, -66], [0, -159, 0, 0], [-345, 0, 0, 0], [0, 165, 0, 0], [0, 0, 229, 0], [0, 0, 0, -97], [0, 0, 0, -10], [0, 0, 555, 0], [0, 0, 0, -45], [0, -336, 0, 0], [0, 0, 873, 0], [0, 0, -250, 0], [0, 0, 481, 0], [0, 731, 0, 0], [0, 0, 0, 173], [-763, 0, 0, 0], [-1500, 0, 0, 0], [0, 0, 0, 10], [0, -765, 0, 0], [-315, 0, 0, 0], [0, 860, 0, 0], [-1563, 0, 0, 0], [0, 0, 0, 65], [0, 0, 415, 0], [0, 0, 0, -465], [0, 282, 0, 0], [0, 0, 625, 0], [0, 0, 0, 325], [0, 0, -1636, 0], [0, 0, 0, 1065], [-1955, 0, 0, 0], [3180, 0, 0, 0], [0, -1265, 0, 0], [0, 0, 0, 830], [0, -1868, 0, 0], [0, -849, 0, 0], [0, 0, -2805, 0], [1365, 0, 0, 0], [0, 0, 0, -280], [0, 0, -3285, 0], [0, 450, 0, 0], [0, 0, 695, 0], [0, 0, 0, -1301], [0, 0, 2203, 0], [0, -1825, 0, 0], [2515, 0, 0, 0], [-5208, 0, 0, 0], [0, 2683, 0, 0], [0, 0, 180, 0], [0, 148, 0, 0], [0, -915, 0, 0], [0, 0, -3575, 0], [5595, 0, 0, 0], [0, 0, 0, 1675], [-3220, 0, 0, 0], [0, 0, 0, 1527], [0, 0, -2322, 0], [0, 0, -3715, 0], [0, 0, 0, -2375], [0, 1273, 0, 0], [4835, 0, 0, 0], [0, 0, 0, 860], [0, 3201, 0, 0], [-6951, 0, 0, 0], [0, 0, -2585, 0], [6909, 0, 0, 0], [0, 0, -3883, 0], [0, 2460, 0, 0], [7576, 0, 0, 0], [0, 0, 0, -3575], [0, 0, 9210, 0], [0, 145, 0, 0], [0, 0, 0, 2369], [-5868, 0, 0, 0], [6935, 0, 0, 0], [0, 0, 1710, 0], [0, 0, 0, -2434], [0, 3267, 0, 0], [0, -6135, 0, 0], [0, 0, 0, 835], [0, 0, 0, 1940], [0, 0, 0, -665], [0, -2916, 0, 0], [0, 0, -9277, 0], [0, 0, 7695, 0], [0, 293, 0, 0], [-359, 0, 0, 0], [0, 0, 0, -5452], [-14775, 0, 0, 0], [0, -1011, 0, 0], [-24381, 0, 0, 0], [0, 0, 0, -1075], [0, 0, 635, 0], [0, 3804, 0, 0], [0, 0, 11835, 0], [0, 0, 0, -2645], [0, 0, 1430, 0], [0, 1957, 0, 0], [12845, 0, 0, 0], [-24360, 0, 0, 0], [0, -175, 0, 0], [0, 0, 0, -510], [-11109, 0, 0, 0], [0, 1048, 0, 0], [0, 6117, 0, 0], [-15225, 0, 0, 0], [0, 0, 0, -290], [0, 0, 3831, 0], [-21980, 0, 0, 0], [0, 0, 555, 0], [0, 0, 0, 769], [0, 5855, 0, 0], [22974, 0, 0, 0], [-10297, 0, 0, 0], [0, 8055, 0, 0], [0, -4072, 0, 0], [0, 0, 3425, 0], [0, 0, 0, 6925], [0, 0, 0, -2526], [0, 0, 0, -6575], [11900, 0, 0, 0], [0, 0, -6785, 0], [0, 0, -13610, 0], [0, 0, 22325, 0], [0, 4037, 0, 0], [0, 0, 13970, 0], [-18597, 0, 0, 0], [0, 1640, 0, 0], [4713, 0, 0, 0], [0, 0, 0, 3059], [0, 0, 0, -7662], [0, 0, -5735, 0], [0, 9300, 0, 0], [0, 0, -7950, 0], [0, 0, 22615, 0], [0, -4175, 0, 0], [0, 0, 0, -1209], [-16555, 0, 0, 0], [0, 0, 5024, 0], [0, 0, 0, 5290], [0, 6357, 0, 0], [0, 0, 0, -5133], [0, 0, 0, -1710], [0, 0, 0, 785], [0, 0, -25307, 0], [0, 0, 8080, 0], [0, 0, 6375, 0], [0, 0, 0, 8047], [-16980, 0, 0, 0], [0, 0, 0, 1742], [0, 11235, 0, 0], [-43155, 0, 0, 0], [0, -18729, 0, 0], [0, 0, 0, 995], [0, 0, -24085, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_784_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_784_4_f_f();".
// 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_4_f_f(:prec:=4)
    chi := MakeCharacter_784_f();
    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_784_4_f_f();".
// 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_4_f_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-7, 0, 1]>],Snew);
    return Vf;
end function;