
// Make newform 19.6.a.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_19_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_19_6_a_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_19_6_a_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_19_a();"
function MakeCharacter_19_a()
    N := 19;
    order := 1;
    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;

function MakeCharacter_19_a_Hecke(Kf)
    return MakeCharacter_19_a();
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 := 6;
    raw_aps := [[-6], [4], [54], [248], [204], [-370], [1554], [361], [-408], [6174], [-7840], [-5146], [-7830], [-12532], [2592], [-20778], [18972], [-18418], [-11548], [-72984], [59114], [-44752], [-27660], [20730], [14018], [145782], [90152], [36108], [32654], [-145806], [-259936], [-124092], [84330], [-155476], [-232794], [496184], [128078], [-570652], [-39864], [223086], [316980], [857270], [783600], [231074], [438966], [-385000], [438740], [-372352], [1011012], [571382], [-594678], [-380544], [-1076782], [-113316], [553218], [824088], [1521582], [-1083040], [658598], [356346], [405116], [-948570], [-1162348], [-2557176], [-1059430], [2644254], [2390828], [-1123822], [2003868], [3608078], [-1828254], [-2610120], [391136], [262070], [824060], [-3897792], [1974006], [-2844034], [5802354], [5353370], [-1141980], [-740794], [1303536], [-7235566], [-415336], [3932028], [-4766238], [2722154], [2466750], [-1637344], [5945652], [-3855312], [-4548472], [1668204], [-956044], [5704104], [6661086], [5915466], [5568716], [744878], [-6024028], [-5297442], [-6333324], [-3075078], [359036], [6431042], [-7473300], [-9414510], [-1223400], [4642682], [-2066464], [-16040842], [8109834], [6431468], [5742920], [6627810], [-16711228], [-599400], [11243262], [10089108], [22193558], [10078466], [-6393882], [-22539636], [-2240428], [7241838], [-11047594], [-17640768], [10222376], [-22706002], [13981700], [21238344], [1080080], [-6160858], [-6964710], [10834946], [26279814], [-15438988], [11925534], [34937802], [-5936020], [20835846], [-23726584], [-11161284], [-8580706], [10929288], [38496326], [-6860454], [-6588676], [-16686144], [33425486], [-2043342], [42570452], [-14705832], [-38973172], [34753200], [-18125080], [-4550622], [5078090], [14079534], [20967348], [-1797510], [-23080744], [-55269588], [-30611022], [15546648], [-7445536], [-27192586]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_19_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_19_6_a_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_19_6_a_a(:prec:=1)
    chi := MakeCharacter_19_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_19_6_a_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_19_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_19_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![6, 1]>],Snew);
    return Vf;
end function;