
// Make newform 83.13.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_83_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_83_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_83_13_b_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_83_13_b_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_83_b();"
function MakeCharacter_83_b()
    N := 83;
    order := 2;
    char_gens := [2];
    v := [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_83_b_Hecke();"
function MakeCharacter_83_b_Hecke(Kf)
    N := 83;
    order := 2;
    char_gens := [2];
    char_values := [[-1]];
    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 := 13;
    raw_aps := [[0], [-617], [0], [-231577], [-3198553], [0], [-30626737], [0], [-231011422], [991523567], [-101624713], [-5012228257], [-9474786718], [0], [0], [0], [-48671434393], [86434023647], [0], [0], [0], [0], [326940373369], [0], [0], [0], [0], [0], [1153705829807], [-2028122937457], [-6152356290697], [2003395005362], [0], [0], [0], [-14748738568873], [0], [0], [28487491052663], [-9259362507697], [0], [0], [-38444208821593], [-58405186312702], [116896575833183], [-123112933322398], [0], [0], [16846458044978], [166485137369042], [0], [0], [-240832647030193], [0], [0], [0], [0], [0], [483137365180178], [0], [0], [-1260212655757102], [0], [0], [-361810198208257], [1976646856127663], [0], [0], [0], [3610778436316127], [-1112205603896017], [-3678437311948393], [0], [-2314391137297], [0], [-3433472172140137], [0], [-7340951071684942], [-7698267117575998], [-9358381801203793], [4379728452378887], [0], [-6484433774079913], [0], [0], [-10481459924301577], [0], [0], [0], [18137693505343943], [0], [3282784711836167], [0], [0], [30465083063153927], [0], [4130694618683807], [28177347340931042], [32095004549740103], [0], [-51883675303175017], [-23861088429860302], [5024401215360743], [0], [0], [0], [0], [10847649860824223], [0], [0], [71648154976418498], [0], [-82777400274097537], [4041157924068887], [0], [0], [0], [0], [0], [-57523083368904718], [0], [-46802453982760897], [0], [0], [69028848685033607], [235083154357321202], [0], [0], [254140052624800103], [292603336061366063], [325530944132854322], [0], [229571139496204802], [288508349917631423], [0], [0], [-281962373462031022], [467726649839141618], [0], [0], [-547610081219596078], [0], [0], [0], [0], [-599895076177225753], [-746509860899201617], [-791176768303736977], [795513908172913607], [225912693943000343], [0], [2398261976781887], [0], [0], [696292773401004023], [1143241217201842247], [0], [1282180259869215167], [0], [1171398668397208607], [0], [-159138880792006417], [0], [0], [1612259200433620703], [-1664569617641741662], [-1817715869542443193], [1049547630090462383]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_83_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_83_13_b_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_83_13_b_a(:prec:=1)
    chi := MakeCharacter_83_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 13));
    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_83_13_b_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_83_13_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_83_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,13,sign)));
    Vf := Kernel([<2,R![0, 1]>,<3,R![617, 1]>],Snew);
    return Vf;
end function;