
// Make newform 224.2.u.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_u_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_2_u_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_2_u_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, 0, 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_224_u();"
function MakeCharacter_224_u()
    N := 224;
    order := 8;
    char_gens := [127, 197, 129];
    v := [8, 3, 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_224_u_Hecke();"
function MakeCharacter_224_u_Hecke(Kf)
    N := 224;
    order := 8;
    char_gens := [127, 197, 129];
    char_values := [[1, 0, 0, 0], [0, 1, 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 := 2;
    raw_aps := [[-1, 0, 1, 0], [-1, -1, -1, -1], [-2, -2, 0, 0], [0, 1, 0, 0], [-3, 3, 2, 2], [0, 2, -2, 0], [0, 0, 0, 0], [-1, 3, -3, 1], [-1, 0, 1, 0], [-4, -3, -3, -4], [-6, -2, 0, 2], [-4, -4, 3, -3], [-2, 0, 2, 2], [-4, 4, 1, 1], [0, 2, 6, 2], [5, -5, 2, 2], [-5, -5, 5, -5], [-4, 0, 0, -4], [4, 5, 5, 4], [0, -6, 0, 0], [6, 0, -6, 4], [0, -7, 0, -7], [-1, 7, -7, 1], [-8, 4, -8, 0], [4, -2, 0, 2], [-2, -2, 0, 0], [-6, -6, -6, 0], [4, -4, 5, 5], [-5, 0, 0, 5], [0, -1, -16, -1], [-2, 12, 0, -12], [1, 3, 3, 1], [1, 0, -1, -20], [-1, 1, -5, -5], [-5, 5, -2, -2], [-3, 0, 3, 0], [6, -6, -6, 6], [7, 0, 0, 7], [10, -6, 10, 0], [0, 6, -6, 0], [-7, 2, -2, 7], [-4, 4, 6, 6], [-4, 5, 0, -5], [0, 1, 0, -1], [2, 2, 13, -13], [8, -10, 8, 0], [-2, -1, 1, 2], [-12, -8, 0, 8], [-11, -11, -11, -11], [8, 8, -8, 8], [1, 0, -1, 4], [0, 0, -6, 0], [0, 14, 4, 14], [-3, -3, 11, -11], [-14, -10, 0, 10], [-4, -8, -4, 0], [-10, 4, -4, 10], [0, 6, -20, 6], [-2, 2, 11, 11], [-7, 8, -7, 0], [3, 3, 11, -11], [12, 12, -8, 8], [-15, -5, 5, 15], [8, 0, -8, -16], [-12, 10, -12, 0], [-2, 7, 7, -2], [3, -3, -6, -6], [0, -9, 20, -9], [0, 0, -3, 3], [8, -8, -8, 8], [2, 18, 0, -18], [-5, 24, -5, 0], [0, -6, -4, -6], [18, -18, -3, -3], [-3, -3, -20, 20], [-10, -2, 0, 2], [0, 0, 17, -17], [4, -12, 12, -4], [0, -3, -28, -3], [20, -6, 20, 0], [-3, 3, 3, -3], [13, 13, 2, -2], [0, -4, -16, -4], [0, -20, 4, -20], [-18, 0, 18, 0], [5, 5, 8, -8], [-8, -18, 0, 18], [8, 0, -8, 0], [0, -12, 12, 0], [0, 17, 4, 17], [9, -17, 17, -9], [-18, -8, 0, 8], [-15, 20, -15, 0], [10, -10, -5, -5], [11, -6, 6, -11], [8, 0, -8, 4], [-8, 8, 8, -8], [-4, 0, 4, 18], [-1, 1, 1, 1], [7, 4, 4, 7], [1, -10, -10, 1], [11, 2, -2, -11], [3, 13, -13, -3], [0, 28, 0, 0], [-10, -10, 13, -13], [-10, 10, 0, -10], [-19, 19, 13, 13], [0, -2, 26, -2], [-8, 0, 8, 16], [-10, 16, -10, 0], [-32, 4, 0, -4], [-9, -9, -26, 26], [10, 0, -10, 24], [-7, 7, -7, -7], [-16, 0, 16, -6], [0, 27, 0, -27], [-3, 19, 19, -3], [4, 22, 4, 0], [-1, -10, 10, 1], [-12, -7, 7, 12], [-14, 14, -24, -24], [0, 18, 0, -18], [22, 22, 10, -10], [-10, 10, 1, 1], [7, -21, 21, -7], [-11, 0, 0, -11], [-13, -13, 18, -18], [0, -8, 22, -8], [-2, 0, 2, 30], [-18, 2, 2, -18], [-7, -20, -20, -7], [7, -12, 7, 0], [0, 20, -2, 20], [7, -7, -2, -2], [-6, -2, -6, 0], [12, 24, 0, -24], [2, 2, -6, 6], [1, 5, -5, -1], [-4, -26, -26, -4], [0, 0, 0, 36], [1, -1, -27, -27], [-2, 2, -13, -13], [-4, 0, 4, -44], [1, 1, 4, -4], [-4, -14, -14, -4], [30, -2, 30, 0], [-32, 32, 16, 16], [-14, -26, -14, 0], [15, 15, -17, 17], [0, -21, 0, 21], [6, -19, 19, -6], [0, 18, -2, 18], [-8, -25, 25, 8], [28, 0, -28, 18], [-1, 1, -10, -10], [0, 8, 30, 8], [-16, 0, 16, 10], [8, -24, 0, 24], [6, 0, -6, 20], [-16, -8, 8, 16], [14, 21, -21, -14], [11, 24, 11, 0], [13, 8, 13, 0], [-11, 11, -11, -11], [0, -2, -56, -2], [-2, 0, 2, 2], [12, -27, 0, 27], [-2, -2, 20, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_2_u_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_2_u_a(:prec:=4)
    chi := MakeCharacter_224_u();
    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_224_2_u_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_2_u_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![8, 16, 12, 4, 1]>],Snew);
    return Vf;
end function;