
// Make newform 300.3.f.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_300_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_300_f_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_3_f_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_3_f_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;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 2, 0], [0, 2, 0, -1]];
    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_300_f();"
function MakeCharacter_300_f()
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    v := [1, 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_300_f_Hecke();"
function MakeCharacter_300_f_Hecke(Kf)
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    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, 1, 0, 1], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 4], [0, 0, -4, 0], [0, 2, 0, 0], [0, -10, 0, 0], [0, 0, 12, 0], [0, 0, 0, -16], [26, 0, 0, 0], [0, 0, -4, 0], [0, -26, 0, 0], [58, 0, 0, 0], [0, 0, 0, -28], [0, 0, 0, -40], [0, -74, 0, 0], [0, 0, -52, 0], [26, 0, 0, 0], [0, 0, 0, -4], [0, 0, 0, 0], [0, -46, 0, 0], [0, 0, 68, 0], [0, 0, 0, 28], [-82, 0, 0, 0], [0, -2, 0, 0], [-74, 0, 0, 0], [0, 0, 0, -44], [0, 0, 0, -12], [46, 0, 0, 0], [0, -110, 0, 0], [0, 0, 0, 84], [0, 0, -68, 0], [0, -10, 0, 0], [0, 0, 28, 0], [2, 0, 0, 0], [0, 0, 52, 0], [0, 214, 0, 0], [0, 0, 0, 12], [0, 0, 0, -56], [0, 334, 0, 0], [0, 0, -108, 0], [2, 0, 0, 0], [0, 0, 128, 0], [0, 290, 0, 0], [0, 26, 0, 0], [0, 0, 228, 0], [0, 0, 140, 0], [0, 0, 0, -196], [0, 0, 0, 164], [142, 0, 0, 0], [0, 82, 0, 0], [0, 0, 224, 0], [-46, 0, 0, 0], [0, 0, -84, 0], [0, 254, 0, 0], [0, 0, 0, 88], [-262, 0, 0, 0], [0, 0, -12, 0], [0, -290, 0, 0], [226, 0, 0, 0], [0, 0, 0, 172], [0, -362, 0, 0], [0, 0, 0, -84], [0, 0, 136, 0], [0, -478, 0, 0], [0, 170, 0, 0], [0, 0, -236, 0], [0, -338, 0, 0], [0, 0, 0, -116], [-506, 0, 0, 0], [0, 178, 0, 0], [0, 0, 96, 0], [0, 0, 0, -116], [0, -310, 0, 0], [0, 0, -252, 0], [0, 0, 0, -352], [578, 0, 0, 0], [0, -26, 0, 0], [250, 0, 0, 0], [-290, 0, 0, 0], [0, 0, -196, 0], [674, 0, 0, 0], [0, 0, -312, 0], [0, -334, 0, 0], [0, 0, 68, 0], [0, 0, 0, -44], [-394, 0, 0, 0], [0, 478, 0, 0], [142, 0, 0, 0], [0, 0, 0, 364], [0, 0, 0, -12], [0, 0, 424, 0], [0, 0, 0, -60], [0, 0, 532, 0], [0, 0, -44, 0], [0, 0, 0, -336], [842, 0, 0, 0], [-326, 0, 0, 0], [0, 0, 0, 180], [530, 0, 0, 0], [0, 0, 0, 196], [0, -766, 0, 0], [0, 0, 0, 284], [422, 0, 0, 0], [0, 0, 164, 0], [0, 46, 0, 0], [0, 0, 0, -364], [0, 82, 0, 0], [0, 0, 32, 0], [-334, 0, 0, 0], [0, 0, 0, 212], [0, -214, 0, 0], [0, 1118, 0, 0], [0, 0, -388, 0], [0, 0, 84, 0], [10, 0, 0, 0], [0, 0, 0, 700], [0, 0, 0, 192], [0, 670, 0, 0], [0, 0, -476, 0], [-1222, 0, 0, 0], [0, -334, 0, 0], [0, -1006, 0, 0], [0, 0, 0, -108], [0, 0, -572, 0], [-1034, 0, 0, 0], [-530, 0, 0, 0], [0, 0, -408, 0], [0, 0, 0, -140], [0, 194, 0, 0], [0, 0, -780, 0], [0, 0, 0, 392], [0, 0, 380, 0], [0, 1006, 0, 0], [-758, 0, 0, 0], [-2, 0, 0, 0], [0, 262, 0, 0], [0, 0, 0, 836], [0, 866, 0, 0], [-10, 0, 0, 0], [0, 0, 252, 0], [838, 0, 0, 0], [0, 0, 0, 508], [0, 0, 0, 420], [-1298, 0, 0, 0], [0, 0, 112, 0], [0, 506, 0, 0], [0, 998, 0, 0], [0, 0, 292, 0], [0, 0, 0, 96], [0, 646, 0, 0], [898, 0, 0, 0], [0, 0, 0, -420], [0, 0, 0, -488], [0, 0, 0, -788], [0, 0, 224, 0], [0, 0, 348, 0], [-1594, 0, 0, 0], [0, -674, 0, 0], [430, 0, 0, 0], [0, 0, 0, 44], [0, 730, 0, 0], [0, 0, 0, -532], [0, 0, -852, 0], [0, -346, 0, 0], [0, 0, 0, 424], [0, 0, -564, 0], [0, -458, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_300_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_300_3_f_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_3_f_a(:prec:=4)
    chi := MakeCharacter_300_f();
    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_300_3_f_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_3_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_300_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![-48, 0, 1]>],Snew);
    return Vf;
end function;