
// Make newform 22.6.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_22_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_22_6_a_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_22_6_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_22_a();"
function MakeCharacter_22_a()
    N := 22;
    order := 1;
    char_gens := [13];
    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_22_a_Hecke(Kf)
    return MakeCharacter_22_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 := [[-4], [1], [-51], [-166], [-121], [692], [-738], [1424], [-1779], [-2064], [6245], [-14785], [5304], [17798], [-17184], [-30726], [-34989], [-45940], [25343], [13311], [-53260], [77234], [55014], [125415], [-88807], [1482], [-117496], [-79362], [87842], [-47247], [-239416], [-98142], [400137], [205766], [87726], [-432778], [-34075], [45020], [482556], [-766254], [303399], [-285181], [767067], [411668], [-759258], [-46600], [-932428], [169745], [198078], [-849997], [-401832], [855174], [1125464], [-1197513], [37758], [-631254], [-1080342], [-816100], [1688198], [-879042], [1540268], [720840], [-1039048], [-1251348], [-1443361], [-2012079], [2017337], [264122], [-1710492], [218822], [3681915], [1885284], [-3116659], [1394414], [-4260355], [201765], [1948821], [-1468258], [2246178], [-3614878], [-3812388], [1973462], [-2083590], [-72691], [594392], [4566507], [-5443821], [6703124], [-1259940], [-5023081], [-2356605], [-6722580], [1960013], [-579624], [1369052], [1833426], [-1712661], [-789435], [627392], [3208952], [3426584], [10519830], [5472876], [-11778672], [-8356276], [-13775839], [-12709308], [1008252], [10510032], [-199390], [16190], [-11525344], [16997394], [-18487471], [-4542811], [18428631], [9666035], [-4514301], [-5372349], [9879558], [10805201], [11327486], [-12059490], [-5141664], [13124255], [3659562], [10225211], [24168285], [16824641], [-5041684], [-6263746], [3639756], [-18736957], [489242], [14696940], [24207236], [13526010], [14209424], [-7933335], [-10468512], [11914742], [-1861122], [23015285], [-16835148], [-23529859], [29154921], [-9490522], [-1815528], [-10781209], [-28335456], [-26891884], [-19213227], [11593100], [13185678], [2981948], [29657868], [31805678], [-23344446], [20737208], [26919282], [10189563], [10392354], [-8188768], [-17327439], [-438963], [-27912387], [-42684640], [-22104418]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_22_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_22_6_a_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_22_6_a_b(:prec:=1)
    chi := MakeCharacter_22_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_22_6_a_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_22_6_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_22_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-1, 1]>],Snew);
    return Vf;
end function;