
// Make newform 2736.3.o.h in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_o_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_2736_3_o_h();".
// 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_2736_3_o_h();".
// 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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 6]];
    Rf_basisdens := [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_2736_o();"
function MakeCharacter_2736_o()
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    v := [2, 2, 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_2736_o_Hecke();"
function MakeCharacter_2736_o_Hecke(Kf)
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    char_values := [[1, 0], [1, 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, 0], [0, 0], [1, 0], [-5, 0], [5, 0], [0, -2], [25, 0], [-19, 0], [-10, 0], [0, -5], [0, 5], [0, -3], [0, 5], [-5, 0], [5, 0], [0, 3], [0, 10], [95, 0], [0, -13], [0, 0], [-25, 0], [0, -5], [-130, 0], [0, 15], [0, 2], [-50, 0], [0, 2], [0, -12], [0, 15], [0, -13], [0, -27], [-163, 0], [-95, 0], [-125, 0], [-215, 0], [0, 10], [-190, 0], [-110, 0], [0, 7], [0, 22], [0, -15], [0, -30], [293, 0], [0, 7], [70, 0], [-173, 0], [0, -10], [0, 43], [0, 8], [-145, 0], [-335, 0], [197, 0], [0, 35], [173, 0], [0, 8], [-355, 0], [0, 45], [-110, 0], [-265, 0], [0, -50], [-125, 0], [0, -22], [0, -33], [-235, 0], [-310, 0], [0, 22], [0, -35], [0, -62], [125, 0], [23, 0], [-410, 0], [-475, 0], [-230, 0], [0, -8], [0, 30], [0, -17], [553, 0], [335, 0], [0, -25], [0, 85], [62, 0], [0, 35], [0, -60], [0, -27], [0, -95], [365, 0], [0, 90], [-265, 0], [553, 0], [-485, 0], [-115, 0], [-490, 0], [0, 72], [-82, 0], [-485, 0], [-250, 0], [0, 20], [0, 15], [0, -42], [-25, 0], [0, -2], [745, 0], [0, 37], [0, -50], [-1070, 0], [-25, 0], [725, 0], [-650, 0], [0, -35], [0, -100], [0, 32], [1055, 0], [505, 0], [130, 0], [475, 0], [0, 100], [955, 0], [965, 0], [-935, 0], [0, -10], [0, -80], [0, -22], [0, -107], [0, 132], [715, 0], [430, 0], [-382, 0], [-115, 0], [1075, 0], [530, 0], [547, 0], [0, -113], [0, 20], [1055, 0], [-215, 0], [-145, 0], [0, -48], [0, -22], [0, 83], [457, 0], [0, -60], [-167, 0], [1315, 0], [0, -63], [0, 90], [0, -40], [770, 0], [0, 148], [-557, 0], [0, 117], [0, 22], [25, 0], [-965, 0], [0, -92], [0, 37], [0, 110], [538, 0], [742, 0], [335, 0], [0, 50], [-1210, 0], [0, -117], [-350, 0], [0, 30], [0, 47], [0, -82], [0, -45], [-265, 0], [0, 75], [-215, 0], [0, -145], [0, -5], [-730, 0], [-1030, 0], [0, -190], [-575, 0], [-302, 0], [1465, 0], [0, 78], [1535, 0], [355, 0], [0, 80], [0, -62], [0, -138], [485, 0], [193, 0], [0, -102], [0, -2], [0, -125], [398, 0], [0, -213], [2165, 0], [0, 105], [0, -65], [-2050, 0], [0, 212], [-577, 0], [-670, 0], [-410, 0], [1685, 0], [0, -70], [0, 265], [0, 43], [0, 215], [-1330, 0], [1105, 0], [1555, 0], [0, 238], [-1103, 0], [0, -160], [650, 0], [190, 0], [-365, 0], [0, -88], [0, -35], [0, -25], [-1070, 0], [0, -80], [0, -33], [1225, 0], [0, -140], [0, 215], [0, 275], [1915, 0], [0, 148], [-1417, 0], [0, -38], [0, 75], [0, -308], [-427, 0], [-145, 0], [0, -285], [0, -145], [0, 20], [715, 0], [-355, 0], [-505, 0], [-1055, 0], [1790, 0], [0, -195], [0, 308], [2563, 0], [-2150, 0], [0, 120], [0, 143], [-1507, 0], [1915, 0], [0, 55], [0, 200], [605, 0], [-1630, 0], [-2927, 0], [1790, 0], [0, -115], [1345, 0], [1838, 0], [983, 0], [0, -173], [0, -177], [-790, 0], [0, -143], [0, -298], [-1585, 0], [0, -348], [-2375, 0], [0, 175], [0, -165], [-1895, 0], [0, 92], [-1655, 0], [0, 235], [0, -207], [-1705, 0], [-2285, 0], [0, -23], [-1190, 0], [-595, 0], [0, -385], [0, -210], [-715, 0], [0, 268], [-3590, 0], [-2155, 0], [0, 140], [-1205, 0], [-3355, 0], [-1465, 0], [0, -268], [283, 0], [0, 120], [-1202, 0], [245, 0], [0, -382], [0, 220], [0, -172], [1270, 0], [0, 310], [-1535, 0], [0, 135], [-5, 0], [-2665, 0], [0, 42], [2635, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_h();".
// 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_2736_3_o_h(:prec:=2)
    chi := MakeCharacter_2736_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2736_3_o_h();".
// 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_2736_3_o_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![5, 1]>],Snew);
    return Vf;
end function;