
// Make newform 448.2.ba.a in Magma, downloaded from the LMFDB on 17 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_448_ba();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_448_ba_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_2_ba_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_448_2_ba_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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_ba();"
function MakeCharacter_448_ba()
    N := 448;
    order := 12;
    char_gens := [127, 197, 129];
    v := [12, 9, 8];
    // 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_ba_Hecke();"
function MakeCharacter_448_ba_Hecke( : Kf := false)
    N := 448;
    order := 12;
    char_gens := [127, 197, 129];
    char_values := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := 2;
    raw_aps := [[0, 0, 0, 0], [-1, 0, 1, -1], [-1, -1, 2, -1], [0, -2, 0, 3], [3, -2, -1, -1], [3, 2, -2, -3], [0, -2, -3, -2], [-2, 2, -1, -3], [-2, -1, -2, 0], [-1, 2, -2, 1], [0, 1, 2, 1], [1, 1, -6, 5], [-4, 0, 8, -2], [-1, -4, -4, -1], [6, -1, -6, 2], [3, -7, 4, 4], [9, -4, -5, -5], [-1, 1, 4, 3], [-1, 4, 5, -5], [2, 0, -4, 4], [6, 2, -3, -2], [-8, -5, 8, 10], [-3, 4, -4, 3], [-3, 0, -3, 0], [4, -8, 0, 4], [5, 7, -12, -12], [2, 3, 2, 0], [-2, -2, 1, 1], [-7, 1, 8, -8], [2, -4, 0, 2], [-6, -4, 0, 2], [4, -4, 3, 7], [18, 2, -9, -2], [1, 8, 8, 1], [-3, -3, -4, 7], [-12, -1, 6, 1], [-1, -13, -12, 12], [2, -2, -7, -5], [8, 0, -16, -8], [-3, 3, 2, -1], [-7, -4, 3, -3], [-3, 12, 12, -3], [0, 5, 0, -10], [0, 0, 23, 0], [-3, -10, -10, -3], [8, 5, -4, -5], [13, 8, -8, -13], [4, -16, 0, 8], [5, -10, -15, 15], [5, 5, 8, -13], [-3, 6, -3, 0], [12, -4, 0, 2], [0, -4, -7, -4], [-7, 18, 18, -7], [9, 6, -9, -12], [12, -15, -6, 15], [-5, 9, 14, -14], [0, 7, 0, -14], [-9, 9, 0, 0], [-4, 0, 8, -6], [-1, 12, -11, -11], [3, -8, -8, 3], [-9, 0, 0, 9], [-4, -3, 2, 3], [-9, -6, -9, 0], [9, -9, -10, -1], [14, 14, -13, -1], [-20, 16, 0, -8], [17, -6, -11, -11], [1, 14, -14, -1], [0, 12, 7, 12], [2, 1, 2, 0], [0, 7, -6, 7], [1, 1, -2, 1], [19, -14, -14, 19], [-20, -3, 20, 6], [1, 15, -16, -16], [-1, 1, 14, 13], [15, 0, -15, 0], [-10, 0, 5, 0], [19, 0, 0, -19], [-5, 10, 10, -5], [0, 5, 22, 5], [-2, -36, 0, 18], [2, -11, 2, 0], [8, 8, -9, 1], [-6, -20, 0, 10], [-3, 24, -3, 0], [15, 10, -10, -15], [-16, 16, 0, -8], [-4, 4, 15, 11], [0, 9, 0, 9], [-20, 5, 10, -5], [-21, 18, 18, -21], [6, -6, -15, -9], [8, 0, -16, -18], [-11, 11, 8, -3], [-34, 14, 17, -14], [24, 24, -25, 1], [11, -11, 2, 13], [15, 22, -22, -15], [5, 3, -2, 2], [-17, 8, 25, -25], [7, -18, 7, 0], [7, 10, -17, -17], [0, -14, 11, -14], [-11, -8, -8, -11], [1, 6, -1, -12], [-28, 5, 14, -5], [0, 0, 0, -10], [-2, 11, 2, -22], [13, -11, -2, -2], [-2, 0, 4, -4], [-19, 12, 7, 7], [-14, 0, 28, 8], [0, -8, 25, -8], [1, 12, -12, -1], [-40, -1, 20, 1], [17, -17, -22, -5], [-13, -16, 16, 13], [-17, 7, 10, 10], [20, 24, 0, -12], [-17, -17, -22, 39], [-1, -32, 33, 33], [10, -10, -15, -5], [-9, -12, 12, 9], [25, 25, -2, -23], [16, -9, -16, 18], [10, 0, -20, -24], [-21, 21, -12, -33], [5, -30, -35, 35], [-4, 0, 8, -18], [-6, 11, 6, -22], [-5, -10, -10, -5], [9, -2, 9, 0], [-4, 16, 0, -8], [-11, 7, 4, 4], [-1, 10, 11, -11], [-17, 10, -10, 17], [-30, -8, 15, 8], [3, -20, -20, 3], [3, 3, -10, 7], [28, -7, -14, 7], [-29, 24, 24, -29], [13, -7, -20, 20], [-16, 0, 32, -10], [-31, 14, 14, -31], [2, 12, -1, -12], [-14, -14, 25, -11], [-24, -5, 24, 10], [-21, 21, 22, 1], [50, 0, 0, 0], [5, 0, 0, -5], [4, 39, 4, 0], [-7, -10, 17, 17], [10, -20, 0, 10], [10, 1, 10, 0], [-5, -26, 5, 52], [4, 0, -8, 36], [-17, -11, 6, -6], [22, -22, -3, 19], [-2, 0, 4, -20], [14, 0, -28, -36], [38, 38, -37, -1], [0, -8, 31, -8], [-24, 1, 12, -1], [0, 9, -24, 9], [3, -3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_448_ba_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_448_2_ba_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_448_2_ba_a(:prec:=4)
    chi := MakeCharacter_448_ba();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_ba_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_448_2_ba_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_448_ba();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 4, 5, 2, 1]>],Snew);
    return Vf;
end function;