
// Make newform 22.6.a.c 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_c();".
// 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_c();".
// 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], [-29], [-31], [-230], [121], [112], [-1142], [-612], [-1941], [1192], [-1037], [8083], [-10444], [58], [8656], [-20318], [-21351], [47044], [48093], [-24967], [-42288], [-72410], [-15806], [-114761], [-5159], [-61426], [185896], [-23970], [-56326], [-261903], [87404], [265122], [245857], [-363594], [-55750], [65642], [-275367], [291940], [-337344], [-116742], [-19107], [-16177], [685333], [-309292], [-120930], [915536], [-134580], [468839], [275022], [-642281], [-1504848], [-304694], [1271808], [629965], [544086], [-1989334], [1754458], [-1658240], [-42634], [319510], [-2027348], [718844], [-1981420], [-1510300], [2000919], [-259331], [51203], [266870], [622800], [2436490], [1559571], [1919608], [3612251], [3939682], [-2188289], [768387], [324313], [334758], [-902022], [-5004574], [-3001236], [4562238], [4897826], [6728757], [-3350336], [-7124343], [-2709285], [2413612], [-6560652], [-4724207], [-2284439], [951544], [3514843], [-5787192], [1029124], [-727370], [-1946297], [-10313283], [-6868396], [10054476], [9857248], [-14589166], [-10241316], [-751816], [-7018540], [-3363767], [14058548], [-5394196], [-12020376], [1646358], [4884506], [3490112], [9120722], [14663515], [-16387005], [-3268353], [8328425], [2496941], [-789105], [-8313926], [-4335171], [7293134], [15581402], [-2169304], [-13219475], [-25939506], [3575311], [-1958141], [15535975], [-14600204], [-20668246], [11706452], [-12760715], [-14086854], [23382160], [-10957520], [16933594], [-7516552], [3937885], [-17360900], [27085002], [35938390], [-505509], [-29995488], [29694153], [-14137137], [-4685394], [-41284552], [-3548051], [-30760536], [30553528], [42101773], [57164], [-11610630], [20985492], [47412500], [40432586], [33075690], [-31513244], [-9541470], [22420753], [16898478], [-30694648], [33566411], [24789677], [5226059], [24082600], [-13260598]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_22_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![29, 1]>],Snew);
    return Vf;
end function;