
// Make newform 189.3.d.d in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_189_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_189_d_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_189_3_d_d();".
// 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_189_3_d_d();".
// 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 := [961, 0, 434, 0, 165, 0, 14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-2139, 0, -1870, 0, 110, 0, -1, 0], [0, 961, 0, 2310, 0, 165, 0, 14], [0, 28334, 0, 5995, 0, 550, 0, 26], [0, 11191, 0, 2310, 0, 165, 0, 14], [-721, 0, 0, 0, 0, 0, 1, 0], [0, -1922, 0, -3597, 0, -330, 0, -28], [18972, 0, 11990, 0, 1100, 0, 83, 0]];
    Rf_basisdens := [1, 5115, 5115, 5115, 1705, 165, 1023, 5115];
    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_189_d();"
function MakeCharacter_189_d()
    N := 189;
    order := 2;
    char_gens := [29, 136];
    v := [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_189_d_Hecke();"
function MakeCharacter_189_d_Hecke(Kf)
    N := 189;
    order := 2;
    char_gens := [29, 136];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [-2, -1, 0, 0, 0, 1, 0, -1], [0, 0, -1, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, -2], [0, 0, 0, 3, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3], [0, 0, 5, 0, 0, 0, 4, 0], [0, 0, 11, 0, 0, 0, 1, 0], [0, -4, 0, 0, 0, 0, 0, -13], [29, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 5, 4, 0, 0, 0], [-22, 0, 0, 0, 0, 7, 0, 0], [0, 0, 0, 1, -5, 0, 0, 0], [0, 0, 8, 0, 0, 0, -2, 0], [0, 0, 0, -13, -1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 18], [44, 0, 0, 0, 0, 13, 0, 0], [0, 0, -28, 0, 0, 0, 7, 0], [0, 11, 0, 0, 0, 0, 0, -20], [14, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 7, 5, 0, 0, 0], [0, 0, 0, -15, 0, 0, 0, 0], [0, -22, 0, 0, 0, 0, 0, -18], [0, 0, 0, 13, -7, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -17], [0, 0, -1, 0, 0, 0, 1, 0], [-97, 0, 0, 0, 0, -9, 0, 0], [0, 0, 2, 0, 0, 0, 12, 0], [-34, 0, 0, 0, 0, 23, 0, 0], [0, 0, 0, -1, 7, 0, 0, 0], [0, 0, -55, 0, 0, 0, 3, 0], [0, -20, 0, 0, 0, 0, 0, -30], [0, 0, -52, 0, 0, 0, -6, 0], [122, 0, 0, 0, 0, -14, 0, 0], [0, -1, 0, 0, 0, 0, 0, -10], [-34, 0, 0, 0, 0, 41, 0, 0], [0, 0, 0, -34, 10, 0, 0, 0], [0, 0, 0, 10, 11, 0, 0, 0], [0, 0, -7, 0, 0, 0, -19, 0], [0, 34, 0, 0, 0, 0, 0, 86], [0, 0, 35, 0, 0, 0, 24, 0], [-82, 0, 0, 0, 0, -40, 0, 0], [0, 0, -64, 0, 0, 0, 4, 0], [0, 13, 0, 0, 0, 0, 0, 5], [98, 0, 0, 0, 0, -62, 0, 0], [0, -3, 0, 0, 0, 0, 0, 29], [0, 0, 0, 34, -8, 0, 0, 0], [0, 28, 0, 0, 0, 0, 0, 42], [0, 0, 74, 0, 0, 0, 8, 0], [0, 0, -7, 0, 0, 0, -3, 0], [0, -35, 0, 0, 0, 0, 0, 82], [0, 0, 0, 42, 10, 0, 0, 0], [0, 0, 0, 56, 3, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, -6, -11, 0, 0, 0], [0, 68, 0, 0, 0, 0, 0, -20], [-13, 0, 0, 0, 0, -68, 0, 0], [0, 0, 89, 0, 0, 0, 9, 0], [0, -28, 0, 0, 0, 0, 0, -76], [0, 0, 0, 11, -7, 0, 0, 0], [0, -62, 0, 0, 0, 0, 0, -19], [0, 0, 0, 29, -27, 0, 0, 0], [0, 63, 0, 0, 0, 0, 0, -106], [0, 0, -7, 0, 0, 0, -47, 0], [212, 0, 0, 0, 0, 20, 0, 0], [53, 0, 0, 0, 0, -32, 0, 0], [0, 0, 56, 0, 0, 0, -27, 0], [0, -20, 0, 0, 0, 0, 0, 40], [0, 0, 0, -17, 12, 0, 0, 0], [0, 0, -19, 0, 0, 0, -11, 0], [0, 65, 0, 0, 0, 0, 0, 57], [-199, 0, 0, 0, 0, 85, 0, 0], [290, 0, 0, 0, 0, 71, 0, 0], [0, 0, 0, 26, -24, 0, 0, 0], [0, 0, 74, 0, 0, 0, 40, 0], [0, 14, 0, 0, 0, 0, 0, -68], [0, 0, -7, 0, 0, 0, 9, 0], [0, 71, 0, 0, 0, 0, 0, 182], [0, 0, 0, 8, 14, 0, 0, 0], [-481, 0, 0, 0, 0, -62, 0, 0], [0, 0, -187, 0, 0, 0, -10, 0], [0, -47, 0, 0, 0, 0, 0, -46], [0, 20, 0, 0, 0, 0, 0, -54], [0, 0, -115, 0, 0, 0, 22, 0], [0, 0, -28, 0, 0, 0, -44, 0], [-103, 0, 0, 0, 0, -38, 0, 0], [0, 0, 0, 4, 43, 0, 0, 0], [692, 0, 0, 0, 0, -26, 0, 0], [0, 0, 0, -51, -27, 0, 0, 0], [0, 0, 0, -38, 6, 0, 0, 0], [-304, 0, 0, 0, 0, -49, 0, 0], [0, 0, -4, 0, 0, 0, 1, 0], [-244, 0, 0, 0, 0, 53, 0, 0], [0, 0, 0, -57, -33, 0, 0, 0], [0, 0, 0, -85, -9, 0, 0, 0], [0, 0, 0, 78, -23, 0, 0, 0], [0, 107, 0, 0, 0, 0, 0, -155], [29, 0, 0, 0, 0, 74, 0, 0], [-4, 0, 0, 0, 0, 160, 0, 0], [0, 0, -298, 0, 0, 0, 10, 0], [0, 0, 0, -61, 29, 0, 0, 0], [0, 0, -28, 0, 0, 0, 60, 0], [314, 0, 0, 0, 0, -40, 0, 0], [0, -90, 0, 0, 0, 0, 0, -60], [0, 0, 0, 88, -30, 0, 0, 0], [0, 0, 0, -72, 15, 0, 0, 0], [0, 0, 263, 0, 0, 0, -18, 0], [0, 35, 0, 0, 0, 0, 0, -24], [0, 32, 0, 0, 0, 0, 0, 44], [-529, 0, 0, 0, 0, -61, 0, 0], [0, 0, 149, 0, 0, 0, 85, 0], [0, 183, 0, 0, 0, 0, 0, -79], [-52, 0, 0, 0, 0, -171, 0, 0], [0, 0, 296, 0, 0, 0, -30, 0], [0, -62, 0, 0, 0, 0, 0, -63], [0, 0, 0, -33, 55, 0, 0, 0], [0, 0, 47, 0, 0, 0, -87, 0], [0, 0, -91, 0, 0, 0, 66, 0], [0, 37, 0, 0, 0, 0, 0, -12], [-418, 0, 0, 0, 0, 10, 0, 0], [0, 0, 0, 151, -4, 0, 0, 0], [0, 0, -325, 0, 0, 0, 4, 0], [0, -10, 0, 0, 0, 0, 0, -102], [0, 0, 179, 0, 0, 0, 115, 0], [581, 0, 0, 0, 0, -77, 0, 0], [0, 0, 0, 153, 19, 0, 0, 0], [0, -70, 0, 0, 0, 0, 0, 222], [0, 207, 0, 0, 0, 0, 0, 88], [-574, 0, 0, 0, 0, -58, 0, 0], [0, 0, 110, 0, 0, 0, -9, 0], [-298, 0, 0, 0, 0, -203, 0, 0], [578, 0, 0, 0, 0, -18, 0, 0], [0, 0, 0, -43, 19, 0, 0, 0], [0, 229, 0, 0, 0, 0, 0, 48], [0, 0, 0, -63, 64, 0, 0, 0], [0, 30, 0, 0, 0, 0, 0, -158], [0, 0, 0, -139, -32, 0, 0, 0], [0, 0, -217, 0, 0, 0, 13, 0], [0, -295, 0, 0, 0, 0, 0, 217], [0, 0, -271, 0, 0, 0, 3, 0], [-64, 0, 0, 0, 0, -154, 0, 0], [0, 0, -262, 0, 0, 0, -41, 0], [0, 127, 0, 0, 0, 0, 0, 38], [0, 0, 0, 85, -29, 0, 0, 0], [0, -130, 0, 0, 0, 0, 0, -162], [0, 0, 0, -35, -8, 0, 0, 0], [0, -152, 0, 0, 0, 0, 0, 243], [0, 0, 125, 0, 0, 0, 49, 0], [584, 0, 0, 0, 0, -26, 0, 0], [0, 0, 0, 237, -40, 0, 0, 0], [578, 0, 0, 0, 0, 44, 0, 0], [0, 0, 0, -253, 61, 0, 0, 0], [572, 0, 0, 0, 0, 100, 0, 0], [0, 0, -205, 0, 0, 0, 25, 0], [-322, 0, 0, 0, 0, -250, 0, 0], [0, 0, 0, 151, 37, 0, 0, 0], [0, 108, 0, 0, 0, 0, 0, -354], [0, 0, 0, 47, 38, 0, 0, 0], [0, 0, -304, 0, 0, 0, -71, 0], [0, 0, -43, 0, 0, 0, -127, 0], [-604, 0, 0, 0, 0, -214, 0, 0], [0, 0, 0, 141, 5, 0, 0, 0], [0, 0, -76, 0, 0, 0, -80, 0], [0, 0, 0, -65, 67, 0, 0, 0], [-772, 0, 0, 0, 0, 23, 0, 0], [0, -82, 0, 0, 0, 0, 0, -324]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_189_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_189_3_d_d();".
// 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_189_3_d_d(:prec:=8)
    chi := MakeCharacter_189_d();
    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_189_3_d_d();".
// 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_189_3_d_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_189_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![31, 0, -14, 0, 1]>],Snew);
    return Vf;
end function;