
// Make newform 21.8.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_21_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_21_8_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_21_8_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_21_a();"
function MakeCharacter_21_a()
    N := 21;
    order := 1;
    char_gens := [8, 10];
    v := [1, 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_21_a_Hecke(Kf)
    return MakeCharacter_21_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 := 8;
    raw_aps := [[2], [27], [-278], [-343], [-4496], [-7274], [11382], [-15884], [86100], [40702], [-44760], [-580962], [-171658], [-741148], [1071720], [-1721778], [-1557012], [2597998], [-963548], [-4063380], [-5370222], [4094936], [-1343124], [9081574], [6487914], [18223154], [743456], [8068192], [-11964762], [1429522], [16452120], [-23974916], [-3200958], [-20394252], [23230734], [286088], [63986734], [-80815052], [-106077104], [-58686790], [-119057776], [32084494], [58111252], [68612114], [111444286], [-176922984], [145063804], [216946672], [257152164], [45393710], [-54777822], [-412963524], [-79535638], [-10578980], [125047358], [-525670988], [420766914], [-101338904], [283426366], [-591103342], [-100607300], [58515218], [-235448588], [-441119904], [-564265078], [63721206], [-67071484], [653040898], [542119736], [-1127829458], [1348436230], [180002108], [1324093920], [-621113890], [1391155188], [278241360], [-781512906], [-2246957634], [-348218550], [671432162], [-422351148], [-847125202], [82849476], [-680200958], [-1124109336], [-2033393448], [564995490], [2009345530], [2491429130], [788855512], [-1199494092], [-3062539144], [-696198584], [-3190012840], [-3410050964], [5448829848], [4080250130], [-2913219762], [-1109980700], [5383341686], [-6386297708], [2699208454], [-3490389212], [5152618690], [-3060512404], [3451067602], [-5672141876], [4061241078], [2755435212], [-3948314886], [-3860423968], [8883400798], [4680372642], [-1960997660], [6874556480], [-7527797382], [2932450820], [-6566420800], [-6468648546], [8043128808], [-7047259178], [2718880018], [8168790786], [-6839263464], [-5426135860], [3598860510], [-6268473426], [9102382608], [-12544499920], [-399754746], [-3966978156], [-10611199620], [6590693248], [235834430], [-3072826842], [-15246609742], [12484888898], [-6266740772], [26087173162], [2649458058], [9947611124], [-22091544802], [-4400683392], [-10006245216], [-18602904634], [-1520963792], [19441006550], [3662977798], [-21054876964], [31725577452], [-644795882], [-19548055866], [-1350384916], [27414891104], [-19963456732], [9435444036], [-8841711688], [32156362574], [7272869946], [-15092806942], [42137545832], [33712013162], [9549910480], [-6182337708], [-20933916630], [2716344168], [-25184001616], [26284214182]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_21_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_21_8_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_21_8_a_a(:prec:=1)
    chi := MakeCharacter_21_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_21_8_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_21_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_21_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![-2, 1]>],Snew);
    return Vf;
end function;