
// Make newform 448.3.c.f in Magma, downloaded from the LMFDB on 22 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_448_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_448_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_448_3_c_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_448_3_c_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 := [2, 0, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [4, 0, 2, 0], [0, 12, 0, 4]];
    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_448_c();"
function MakeCharacter_448_c()
    N := 448;
    order := 2;
    char_gens := [127, 197, 129];
    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_448_c_Hecke();"
function MakeCharacter_448_c_Hecke(Kf)
    N := 448;
    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], [0, 1, 0, 1], [1, -1, 1, 1], [-2, 0, 4, 0], [0, -5, 0, 1], [0, 0, 0, 2], [0, -7, 0, -2], [-6, 0, 8, 0], [-26, 0, 4, 0], [0, 4, 0, -2], [-10, 0, -12, 0], [0, -14, 0, 6], [46, 0, 0, 0], [0, -8, 0, -10], [38, 0, 8, 0], [0, -9, 0, -6], [0, -21, 0, 3], [-2, 0, 8, 0], [-62, 0, -6, 0], [0, 14, 0, 0], [-62, 0, -10, 0], [0, -11, 0, 12], [0, 18, 0, -4], [0, -36, 0, -14], [0, 21, 0, -17], [0, -6, 0, 4], [-82, 0, 24, 0], [22, 0, -16, 0], [42, 0, -10, 0], [42, 0, 48, 0], [0, -23, 0, 20], [-14, 0, 48, 0], [0, 11, 0, -26], [102, 0, 48, 0], [122, 0, -8, 0], [0, 39, 0, 5], [-114, 0, -20, 0], [0, -62, 0, 0], [0, 11, 0, -27], [238, 0, -12, 0], [0, 29, 0, 25], [-30, 0, 14, 0], [138, 0, -50, 0], [22, 0, 44, 0], [0, -14, 0, 36], [-130, 0, -64, 0], [0, 40, 0, -24], [0, 29, 0, 2], [0, -27, 0, 23], [-270, 0, 64, 0], [106, 0, -40, 0], [0, -40, 0, 58], [0, -77, 0, -8], [0, 40, 0, -12], [162, 0, -30, 0], [0, -21, 0, 9], [0, -36, 0, -20], [-74, 0, 116, 0], [50, 0, -96, 0], [0, -85, 0, 56], [0, 149, 0, 23], [0, 13, 0, 20], [0, -118, 0, -14], [0, -2, 0, 70], [38, 0, -20, 0], [-2, 0, -68, 0], [-278, 0, 54, 0], [206, 0, -16, 0], [0, 27, 0, 7], [0, -8, 0, -24], [26, 0, -184, 0], [0, 180, 0, 60], [150, 0, -100, 0], [94, 0, -148, 0], [0, -188, 0, -14], [150, 0, -132, 0], [0, -25, 0, -25], [74, 0, 22, 0], [0, -178, 0, -54], [0, 65, 0, -12], [-218, 0, -72, 0], [522, 0, 96, 0], [0, 176, 0, -2], [0, 2, 0, -114], [-514, 0, -92, 0], [322, 0, 152, 0], [-54, 0, -70, 0], [0, 79, 0, -29], [-30, 0, 70, 0], [0, -151, 0, -18], [0, 36, 0, -26], [218, 0, 120, 0], [334, 0, 72, 0], [238, 0, 140, 0], [0, 66, 0, -26], [0, 83, 0, -65], [0, 282, 0, 70], [0, -37, 0, -82], [-778, 0, -88, 0], [270, 0, -4, 0], [470, 0, 96, 0], [0, -15, 0, 14], [-726, 0, -6, 0], [-530, 0, -24, 0], [0, 56, 0, -124], [0, 35, 0, -102], [0, -60, 0, -44], [354, 0, -46, 0], [0, -278, 0, -40], [0, -248, 0, 8], [-218, 0, 8, 0], [810, 0, 10, 0], [0, 47, 0, 60], [258, 0, -230, 0], [-310, 0, -178, 0], [0, -3, 0, -122], [0, -94, 0, 4], [150, 0, -200, 0], [-130, 0, -184, 0], [0, 17, 0, 135], [-350, 0, -296, 0], [0, -191, 0, -83], [270, 0, 16, 0], [0, 45, 0, 36], [566, 0, -72, 0], [326, 0, 280, 0], [0, 336, 0, 62], [0, -26, 0, -84], [0, -357, 0, 39], [-402, 0, 260, 0], [-678, 0, 24, 0], [-86, 0, 24, 0], [6, 0, -40, 0], [0, -234, 0, 170], [0, 76, 0, 110], [0, 129, 0, 97], [0, 121, 0, 78], [0, -17, 0, 57], [-182, 0, -246, 0], [0, 23, 0, -120], [502, 0, -332, 0], [-1598, 0, 2, 0], [-1058, 0, -44, 0], [0, -197, 0, -101], [0, -286, 0, -20], [0, -159, 0, -89], [0, -42, 0, -122], [0, -357, 0, 40], [1186, 0, 110, 0], [-922, 0, 276, 0], [0, -100, 0, 212], [-130, 0, 536, 0], [0, -130, 0, 68], [-546, 0, -268, 0], [-534, 0, 24, 0], [-30, 0, -406, 0], [0, 180, 0, 86], [0, 270, 0, -196], [0, -349, 0, 81], [94, 0, -80, 0], [306, 0, 368, 0], [474, 0, -344, 0], [0, 51, 0, -254], [386, 0, 352, 0], [0, -170, 0, 212], [-318, 0, 478, 0], [0, 145, 0, -83]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_448_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_448_3_c_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_448_3_c_f(:prec:=4)
    chi := MakeCharacter_448_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_448_3_c_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_448_3_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_448_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![32, 0, 16, 0, 1]>,<11,R![-124, 4, 1]>],Snew);
    return Vf;
end function;