
// Make newform 189.3.b.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_189_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_189_b_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_b_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_189_3_b_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 := [9, 0, 8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [4, 0, 1, 0], [0, 6, 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_189_b();"
function MakeCharacter_189_b()
    N := 189;
    order := 2;
    char_gens := [29, 136];
    v := [1, 2];
    // 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_b_Hecke();"
function MakeCharacter_189_b_Hecke(Kf)
    N := 189;
    order := 2;
    char_gens := [29, 136];
    char_values := [[-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, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 1, 0], [0, 3, 0, -4], [-3, 0, 5, 0], [0, 1, 0, -1], [6, 0, -1, 0], [0, -3, 0, 6], [0, 17, 0, 3], [16, 0, 7, 0], [-23, 0, -14, 0], [0, -20, 0, -9], [-7, 0, -15, 0], [0, -11, 0, -1], [0, 28, 0, 2], [0, 41, 0, 1], [24, 0, -34, 0], [39, 0, -1, 0], [0, 42, 0, -9], [49, 0, -23, 0], [-121, 0, -9, 0], [0, -31, 0, 25], [0, -10, 0, -15], [4, 0, -38, 0], [0, -51, 0, 31], [20, 0, 51, 0], [0, -13, 0, 15], [-8, 0, 25, 0], [0, -68, 0, 4], [97, 0, 19, 0], [0, 9, 0, 41], [0, 15, 0, 21], [-50, 0, 10, 0], [0, 84, 0, 0], [134, 0, 6, 0], [165, 0, -19, 0], [-27, 0, 59, 0], [0, -38, 0, -20], [0, 97, 0, 0], [0, 97, 0, -45], [-270, 0, -22, 0], [0, 19, 0, -44], [20, 0, -30, 0], [0, -78, 0, 44], [-59, 0, 76, 0], [-36, 0, 20, 0], [75, 0, 104, 0], [0, -90, 0, -42], [-118, 0, -72, 0], [0, 20, 0, 16], [0, -167, 0, -25], [77, 0, -81, 0], [0, 18, 0, 62], [0, 69, 0, -86], [0, 45, 0, 44], [0, -77, 0, -82], [36, 0, 14, 0], [69, 0, -52, 0], [0, 11, 0, -19], [66, 0, 116, 0], [0, -23, 0, 21], [-414, 0, -13, 0], [0, 107, 0, -71], [-367, 0, 21, 0], [0, -71, 0, 59], [68, 0, 36, 0], [195, 0, -58, 0], [0, 128, 0, -55], [408, 0, -58, 0], [0, -72, 0, -13], [0, -49, 0, 17], [-71, 0, -44, 0], [206, 0, 189, 0], [571, 0, -41, 0], [0, -120, 0, 60], [0, 24, 0, -30], [-374, 0, -104, 0], [0, -123, 0, -39], [305, 0, -117, 0], [0, -164, 0, 56], [179, 0, -210, 0], [0, -137, 0, 32], [-225, 0, -49, 0], [-530, 0, 82, 0], [0, -137, 0, -70], [0, 24, 0, 6], [-183, 0, -106, 0], [0, -7, 0, -118], [90, 0, 92, 0], [0, -199, 0, -11], [0, 156, 0, -32], [135, 0, -211, 0], [0, 256, 0, 93], [243, 0, -223, 0], [0, -75, 0, -165], [0, -309, 0, 61], [0, 307, 0, -106], [177, 0, -172, 0], [137, 0, 90, 0], [192, 0, 32, 0], [0, -170, 0, 52], [0, -301, 0, -69], [0, 16, 0, -14], [-16, 0, -228, 0], [-36, 0, 158, 0], [0, 182, 0, 78], [0, 119, 0, 130], [0, -37, 0, 166], [313, 0, 103, 0], [-10, 0, -72, 0], [-48, 0, 71, 0], [0, 97, 0, -133], [-81, 0, 80, 0], [95, 0, 69, 0], [0, -390, 0, -102], [-418, 0, 75, 0], [0, 133, 0, 93], [0, 57, 0, -27], [0, -343, 0, 142], [459, 0, -43, 0], [570, 0, 212, 0], [0, 154, 0, 71], [0, 253, 0, -196], [206, 0, -102, 0], [0, -33, 0, -201], [-386, 0, 1, 0], [0, 59, 0, -23], [-676, 0, 276, 0], [67, 0, 19, 0], [682, 0, 46, 0], [0, -92, 0, -181], [69, 0, -31, 0], [602, 0, -132, 0], [0, -213, 0, -69], [-607, 0, -177, 0], [0, 388, 0, -105], [654, 0, -274, 0], [0, 202, 0, -167], [0, -449, 0, -19], [203, 0, -408, 0], [0, -359, 0, 309], [88, 0, 34, 0], [0, -330, 0, 69], [335, 0, 225, 0], [0, -181, 0, -129], [-220, 0, 18, 0], [0, -150, 0, -199], [272, 0, 159, 0], [0, 233, 0, 199], [-168, 0, 152, 0], [0, 504, 0, 201], [-352, 0, -402, 0], [0, -333, 0, 211], [1004, 0, -162, 0], [0, 263, 0, -77], [-1114, 0, -72, 0], [0, -95, 0, -63], [544, 0, 238, 0], [0, 452, 0, -187], [0, 112, 0, 63], [0, -147, 0, 13], [-342, 0, 308, 0], [0, -307, 0, -149], [0, -460, 0, -116], [0, -213, 0, -45], [1235, 0, 99, 0], [330, 0, -232, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_189_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_189_3_b_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_189_3_b_b(:prec:=4)
    chi := MakeCharacter_189_b();
    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_b_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_189_3_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_189_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![9, 0, 8, 0, 1]>],Snew);
    return Vf;
end function;