
// Make newform 300.2.m.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_300_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_300_m_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_300_2_m_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_300_2_m_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, -1, 0, 1, -1, 1, 0, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, -1, 0, 0, 0, 0, -1], [0, 1, 0, 0, -1, 0, 1, 0], [-1, 2, 0, 0, 1, -1, 1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 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_300_m();"
function MakeCharacter_300_m()
    N := 300;
    order := 5;
    char_gens := [151, 101, 277];
    v := [5, 5, 4];
    // 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_300_m_Hecke();"
function MakeCharacter_300_m_Hecke(Kf)
    N := 300;
    order := 5;
    char_gens := [151, 101, 277];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 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 := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, -2, -1, -2, -1, -1, 1], [-2, -1, -2, 0, -1, -2, -1, 1], [-1, 0, -1, 0, -1, -1, -1, 1], [-2, 1, -2, -2, -3, -1, 0, 2], [-1, 1, -1, -3, 0, 0, -1, 0], [2, 0, -1, 3, 2, 0, 0, 0], [-1, 3, 0, -3, -2, 2, 0, 0], [0, 0, -3, -3, -6, -2, 3, -1], [0, -2, -2, 1, -2, 0, 2, 0], [0, -3, 0, 1, 1, 0, 0, -1], [1, -2, 1, -2, -5, -3, 0, 2], [-8, -2, -1, 0, -2, -1, 2, 2], [0, 0, 5, -1, 4, 0, 1, -5], [0, 0, 4, -3, 1, 3, 3, -1], [3, -1, 3, 6, 5, -1, 0, -6], [0, -4, 2, 4, -1, -4, 2, -2], [9, -4, 6, 2, 5, 0, 4, 0], [0, 0, 4, -1, 3, -2, 1, 3], [1, 4, 6, -4, 0, 5, 6, -6], [0, 0, -8, 6, -2, -6, -6, 1], [-2, 2, -5, 0, 0, 0, -2, 0], [4, -2, 3, 2, 6, 2, 3, -3], [0, 0, -6, 4, -2, -11, -4, 0], [9, -7, 2, 0, -7, 2, 3, 7], [0, 0, 5, -9, -4, 5, 9, 2], [-12, -2, -7, 0, -2, -7, -4, 2], [-3, 4, -3, -3, -3, 0, 0, 3], [-1, 1, -1, 2, -2, -4, 0, -2], [-2, -5, -3, 5, -9, -7, -3, 3], [-13, 1, -8, -2, -12, 0, -1, 0], [-3, 3, -3, 0, 3, 3, 0, 0], [10, 2, -2, -2, 6, 12, -2, 2], [-7, -10, -3, 0, -10, -3, 1, 10], [5, 4, 3, 0, 4, 3, -2, -4], [9, -2, 7, 0, -2, 7, 3, 2], [-4, -3, -4, -9, -16, -7, 0, 9], [7, -1, -6, -3, 6, 0, 1, 0], [10, -9, -3, 9, 8, 1, -3, 3], [0, 0, -8, -1, -9, -7, 1, -4], [-2, -2, -20, 3, -4, 0, 2, 0], [0, 3, 0, 0, 1, 1, 0, 0], [0, 5, 0, 0, 5, 0, 6, -5], [0, 0, -5, 4, -1, 5, -4, 10], [-5, 1, 9, 0, 1, 9, -4, -1], [3, 10, 7, -10, 12, 13, 7, -7], [-8, 0, -7, 0, -12, -8, -7, 7], [-6, 3, 7, -3, -13, -3, 7, -7], [0, 0, 6, -9, -3, -3, 9, 8], [-5, -3, -19, 4, -8, 0, 3, 0], [2, -1, -1, 1, 16, 1, -1, 1], [2, -2, 2, 15, 11, -4, 0, -15], [3, 9, 3, 0, 9, 3, -7, -9], [-2, 7, 21, 0, 7, 21, 4, -7], [17, 0, 17, 4, 16, 12, 0, -4], [4, -14, 12, -2, -10, 0, 14, 0], [0, 0, 8, 2, 10, 0, -2, -15], [0, -10, -5, 10, -7, -10, -5, 5], [-1, 0, -7, 5, -1, 0, 0, 0], [6, -10, 12, -3, -4, 0, 10, 0], [18, 2, 8, 0, 2, 8, 0, -2], [1, -6, -1, 0, -6, -1, 7, 6], [9, 8, 12, -8, 21, 17, 12, -12], [19, 0, 19, 14, 20, 6, 0, -14], [8, 1, -11, 5, 9, 0, -1, 0], [-24, 13, -25, -8, -11, 0, -13, 0], [-2, 4, -2, -7, 2, 9, 0, 7], [0, 0, 3, 1, 4, 1, -1, 1], [-16, -5, 5, 0, -5, 5, -3, 5], [0, 0, 14, 6, 20, -14, -6, -11], [-13, -3, -13, -16, -28, -12, 0, 16], [-9, 0, -9, 9, -9, 0, 0, 0], [10, -3, -4, 3, 11, 7, -4, 4], [0, 0, -14, 10, -4, -12, -10, 3], [-3, 2, -10, 0, -1, 0, -2, 0], [4, 5, -1, -5, 19, 9, -1, 1], [0, 0, -17, 5, -12, -18, -5, 4], [16, -3, -1, 0, -3, -1, -6, 3], [-2, 1, -2, -7, -5, 2, 0, 7], [-1, 5, 8, -11, 4, 0, -5, 0], [0, 0, -7, -4, -11, 2, 4, 3], [-24, 2, -10, -13, -22, 0, -2, 0], [-7, 7, -10, 8, 0, 0, -7, 0], [7, -4, -8, 4, 11, 3, -8, 8], [3, -8, -13, 0, -8, -13, -14, 8], [23, 7, 16, 0, 7, 16, -5, -7], [-4, -3, 6, 0, -3, 6, -6, 3], [-19, 6, -10, -6, -12, -13, -10, 10], [9, 0, 9, -6, 3, 9, 0, 6], [13, -3, -9, 3, 10, 0, 3, 0], [0, 0, -9, -2, -11, 12, 2, -2], [21, 5, 21, 6, 27, 21, 0, -6], [-16, -6, -16, -5, -20, -15, 0, 5], [0, -4, -8, 0, -4, -8, 6, 4], [0, 0, 5, -12, -7, -5, 12, 4], [-21, -5, -21, -2, -31, -29, 0, 2], [0, 0, 19, -6, 13, 16, 6, -13], [10, -7, 12, 7, -3, 3, 12, -12], [-1, 14, -1, -8, -10, -2, 0, 8], [0, 0, -7, 4, -3, -8, -4, 4], [4, -1, 8, 0, -1, 8, -8, 1], [-2, 0, -2, 0, 17, 17, 0, 0], [-3, 14, -4, 10, 11, 0, -14, 0], [0, 0, -1, 1, 0, -7, -1, 10], [-4, 8, 7, -8, 21, 4, 7, -7], [-11, -16, -11, 8, -13, -21, 0, -8], [16, 5, 15, 0, 5, 15, 18, -5], [18, 5, 20, 0, 5, 20, 13, -5], [2, -12, -1, 0, -12, -1, -4, 12], [-5, 12, 6, 0, 12, 6, 2, -12], [17, 9, 17, 0, 21, 21, 0, 0], [9, 7, -6, -6, 16, 0, -7, 0], [15, -6, 11, 18, 9, 0, 6, 0], [1, -6, 20, 1, -5, 0, 6, 0], [2, -5, 2, 5, -15, -20, 0, -5], [1, 7, 15, 0, 7, 15, 17, -7], [0, 0, 33, -1, 32, 17, 1, -2], [0, 0, 16, -10, 6, 4, 10, 3], [-5, 5, -5, 14, 30, 16, 0, -14], [-4, 4, 12, -4, -20, 0, 12, -12], [9, 9, 3, -9, -10, 18, 3, -3], [-3, -15, 11, 15, 14, -18, 11, -11], [-23, -1, -10, 4, -24, 0, 1, 0], [28, 4, 28, 1, 32, 31, 0, -1], [-14, 2, -8, 0, 2, -8, -2, -2], [-23, -14, -23, -6, -36, -30, 0, 6], [18, -20, 23, 6, -2, 0, 20, 0], [-12, 19, 1, -19, 2, 7, 1, -1], [-4, 15, 10, -2, 11, 0, -15, 0], [17, -15, 17, 15, -1, 2, 17, -17], [6, -4, 7, 0, -4, 7, 12, 4], [0, -18, -9, 0, -18, -9, -13, 18], [-2, 2, -13, 0, 2, -13, -14, -2], [0, -7, -3, 7, 12, -7, -3, 3], [-9, -5, -22, -2, -14, 0, 5, 0], [-18, 17, 0, -17, 7, -1, 0, 0], [-2, 6, -2, -10, 9, 19, 0, 10], [0, 0, -2, 10, 8, 15, -10, -13], [-8, 0, -8, 16, 20, 4, 0, -16], [-12, 2, -23, -2, -20, -10, -23, 23], [0, 0, -14, 7, -7, 13, -7, 13], [-17, 2, -14, -2, -7, -15, -14, 14], [5, -9, -18, 9, -5, -4, -18, 18], [0, 0, 7, -1, 6, -25, 1, 2], [-8, 21, 2, -21, -8, 13, 2, -2], [0, 0, -6, 18, 12, 9, -18, 9], [-11, 1, -11, 0, 1, -11, 12, -1], [24, -4, 24, 7, -3, -10, 0, -7], [-17, 20, -17, -3, 9, 12, 0, 3], [17, 0, 3, 0, 24, 17, 3, -3], [6, -4, 4, -20, 2, 0, 4, 0], [11, 3, -6, -5, 14, 0, -3, 0], [-16, -6, -16, -11, -29, -18, 0, 11], [-13, 13, -8, 0, 13, -8, -6, -13], [19, -17, -3, 17, 0, 2, -3, 3], [17, -10, 8, -6, 7, 0, 10, 0], [0, 0, -4, 22, 18, 16, -22, 7], [-5, -12, -5, -1, 1, 2, 0, 1], [8, 19, 8, 11, 13, 2, 0, -11], [0, 0, 13, -3, 10, 8, 3, 0], [0, 0, 18, -16, 2, 10, 16, 14], [-27, 9, -6, -4, -18, 0, -9, 0], [0, 0, 7, 6, 13, -1, -6, -9], [-18, -10, -3, 10, -8, -28, -3, 3], [13, -11, -24, -14, 2, 0, 11, 0], [2, -10, 2, -18, -15, 3, 0, 18], [0, 0, 11, -8, 3, 12, 8, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_300_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_300_2_m_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_300_2_m_a(:prec:=8)
    chi := MakeCharacter_300_m();
    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_300_2_m_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_300_2_m_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_300_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, -1, -4, 4, 1]>],Snew);
    return Vf;
end function;