
// Make newform 450.2.e.b in Magma, downloaded from the LMFDB on 26 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_450_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_450_e_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_450_2_e_b();".
// 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_450_2_e_b();".
// 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;
    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_450_e();"
function MakeCharacter_450_e()
    N := 450;
    order := 3;
    char_gens := [101, 127];
    v := [2, 3];
    // 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_450_e_Hecke();"
function MakeCharacter_450_e_Hecke( : Kf := false)
    N := 450;
    order := 3;
    char_gens := [101, 127];
    char_values := [[0, -1], [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 := [[-1, 1], [-2, 1], [0, 0], [1, -1], [2, -2], [0, 6], [-2, 0], [6, 0], [0, -1], [-9, 9], [0, 2], [2, 0], [0, 11], [4, -4], [7, -7], [0, 0], [0, 4], [7, -7], [0, 11], [-6, 0], [-4, 0], [12, -12], [11, -11], [1, 0], [8, -8], [-2, 2], [0, 8], [3, 0], [7, 0], [0, -12], [-19, 0], [0, -12], [-12, 12], [0, -16], [0, 1], [-10, 10], [0, -4], [4, 0], [0, 3], [4, -4], [2, 0], [-13, 0], [-6, 6], [0, 10], [8, 0], [-18, 0], [0, -18], [-23, 23], [8, -8], [0, 7], [-10, 0], [0, -28], [-1, 1], [18, 0], [0, -12], [-16, 16], [3, 0], [-14, 0], [22, -22], [3, -3], [0, -1], [0, 18], [-9, 0], [0, -6], [22, -22], [2, -2], [8, -8], [0, -8], [0, 12], [11, -11], [16, -16], [30, 0], [16, -16], [0, -12], [-16, 0], [0, -32], [-19, 19], [-4, 0], [0, 10], [0, -38], [0, -34], [-22, 22], [16, 0], [2, 0], [-24, 24], [9, -9], [18, 0], [10, -10], [21, -21], [0, -36], [36, 0], [-28, 28], [-12, 0], [0, 0], [0, 24], [27, 0], [0, 15], [-37, 0], [-29, 0], [-17, 0], [-35, 35], [-24, 0], [0, 37], [2, -2], [0, 20], [32, 0], [-3, 3], [-30, 0], [0, 12], [22, -22], [0, -1], [-34, 0], [0, 32], [10, -10], [8, 0], [-13, 13], [0, 33], [33, 0], [0, 26], [-20, 20], [0, 10], [-6, 6], [-22, 22], [4, 0], [44, -44], [13, 0], [27, -27], [-44, 0], [-21, 21], [0, -4], [40, 0], [0, 15], [0, -26], [-10, 0], [0, 9], [0, 15], [12, 0], [0, -44], [0, 2], [-30, 0], [-20, 0], [31, -31], [0, -15], [-35, 0], [15, 0], [-38, 38], [-46, 46], [18, -18], [0, -14], [-45, 0], [0, -46], [-15, 0], [3, 0], [0, 0], [-33, 33], [-12, 12], [-36, 0], [18, -18], [52, 0], [0, 41], [51, -51], [2, 0], [0, 17], [-42, 0], [0, -54], [19, -19], [4, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_450_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_450_2_e_b();".
// 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_450_2_e_b(:prec:=2)
    chi := MakeCharacter_450_e();
    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_450_2_e_b();".
// 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_450_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_450_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, -1, 1]>,<11,R![4, -2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;