
// Make newform 126.3.s.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_126_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_126_s_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_126_3_s_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_126_3_s_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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_126_s();"
function MakeCharacter_126_s()
    N := 126;
    order := 6;
    char_gens := [29, 73];
    v := [3, 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_126_s_Hecke();"
function MakeCharacter_126_s_Hecke(Kf)
    N := 126;
    order := 6;
    char_gens := [29, 73];
    char_values := [[-1, 0, 0, 0], [-1, 0, 1, 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, 1, 0, 0], [5, 0, 3, 0], [0, -5, 0, 5], [15, 0, 0, 0], [0, -8, 0, 8], [13, 0, -13, 0], [0, -16, 0, 0], [0, 0, 0, -16], [0, 0, -3, 0], [-17, 0, 17, 0], [0, 0, 0, -57], [-85, 0, 0, 0], [0, 51, 0, 0], [0, 24, 0, -24], [0, -64, 0, 64], [72, 0, -72, 0], [0, 0, -43, 0], [0, 0, 0, 37], [0, 0, 95, 0], [-69, 0, 69, 0], [0, 0, 0, 43], [0, 96, 0, 0], [16, 0, 0, 0], [0, -49, 0, 49], [61, 0, -61, 0], [0, 120, 0, 0], [0, 0, 65, 0], [0, 0, 0, -97], [-171, 0, 0, 0], [0, -83, 0, 0], [0, 152, 0, -152], [-83, 0, 0, 0], [0, -72, 0, 0], [0, 0, -40, 0], [0, 0, -296, 0], [-128, 0, 128, 0], [0, 0, 0, -43], [0, -80, 0, 0], [0, -21, 0, 21], [81, 0, 0, 0], [0, 45, 0, 0], [0, 0, 223, 0], [0, 0, 0, 112], [0, 0, 136, 0], [272, 0, 0, 0], [248, 0, 0, 0], [0, 117, 0, -117], [-433, 0, 433, 0], [0, 161, 0, 0], [0, 0, 0, 243], [0, 0, -158, 0], [0, 0, 0, -168], [0, 225, 0, 0], [0, -120, 0, 120], [0, -343, 0, 343], [-216, 0, 216, 0], [0, 0, -305, 0], [0, 0, 0, 16], [0, 0, -157, 0], [0, 0, 0, 72], [-11, 0, 0, 0], [0, 371, 0, -371], [-553, 0, 553, 0], [0, -96, 0, 0], [61, 0, -61, 0], [135, 0, 0, 0], [0, 72, 0, -72], [-152, 0, 0, 0], [0, 273, 0, -273], [0, -32, 0, 0], [0, 0, -101, 0], [311, 0, -311, 0], [91, 0, 0, 0], [0, -232, 0, 0], [0, -487, 0, 487], [279, 0, -279, 0], [0, 0, 0, 0], [0, 0, 223, 0], [0, 0, 0, 411], [153, 0, 0, 0], [0, -99, 0, 99], [137, 0, 0, 0], [600, 0, -600, 0], [0, -448, 0, 0], [0, 0, 0, -271], [239, 0, -239, 0], [0, 0, 0, 320], [211, 0, 0, 0], [0, -221, 0, 0], [0, 496, 0, -496], [0, 0, -419, 0], [0, 0, 0, -120], [-187, 0, 187, 0], [0, 0, 0, -123], [0, 183, 0, 0], [0, -288, 0, 288], [811, 0, -811, 0], [345, 0, -345, 0], [-864, 0, 0, 0], [0, 671, 0, -671], [0, 523, 0, -523], [0, -543, 0, 0], [0, 0, -477, 0], [0, 0, 263, 0], [0, 0, 0, 376], [0, 567, 0, 0], [0, -552, 0, 552], [383, 0, 0, 0], [765, 0, -765, 0], [0, 0, -408, 0], [0, 0, 0, -647], [0, 0, 947, 0], [-760, 0, 0, 0], [0, 328, 0, -328], [-19, 0, 0, 0], [0, -819, 0, 819], [0, -729, 0, 0], [0, 0, 0, 688], [0, 0, 401, 0], [665, 0, 0, 0], [0, 648, 0, 0], [0, 96, 0, -96], [-221, 0, 221, 0], [0, 0, 0, -136], [328, 0, -328, 0], [0, 717, 0, 0], [1069, 0, 0, 0], [-495, 0, 495, 0], [0, 0, -453, 0], [0, 0, 0, -381], [339, 0, -339, 0], [198, 0, 0, 0], [0, -256, 0, 0], [-929, 0, 0, 0], [0, -121, 0, 121], [0, 0, -90, 0], [0, 0, 0, -816], [0, -121, 0, 121], [-554, 0, 0, 0], [0, -751, 0, 0], [0, 0, -856, 0], [0, 0, 0, 723], [0, 0, -1039, 0], [0, 0, 0, -464], [1463, 0, 0, 0], [0, -48, 0, 48], [1462, 0, -1462, 0], [0, 571, 0, 0], [-1480, 0, 1480, 0], [0, 0, 0, -504], [-115, 0, 0, 0], [0, -843, 0, 0], [0, 0, 501, 0], [0, 0, 0, 144], [653, 0, -653, 0], [0, -9, 0, 0], [761, 0, 0, 0], [0, 200, 0, -200], [0, 763, 0, 0], [0, 0, 0, 760], [1163, 0, 0, 0], [0, -1104, 0, 0], [0, -801, 0, 801], [0, 216, 0, -216], [0, 0, 973, 0], [0, 0, 79, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_126_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_126_3_s_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_126_3_s_b(:prec:=4)
    chi := MakeCharacter_126_s();
    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_126_3_s_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_126_3_s_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_126_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 0, -2, 0, 1]>],Snew);
    return Vf;
end function;