
// Make newform 33.8.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_33_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_33_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_33_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_33_a();"
function MakeCharacter_33_a()
    N := 33;
    order := 1;
    char_gens := [23, 13];
    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_33_a_Hecke(Kf)
    return MakeCharacter_33_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 := [[10], [27], [-410], [-1028], [-1331], [12958], [17062], [-54168], [-11488], [-186654], [-188672], [395886], [-47546], [602088], [-647200], [-1312722], [-2681140], [551190], [459260], [-18072], [-426062], [297764], [5684028], [-6342966], [16651586], [-2083270], [-2390464], [-14061468], [-11132066], [5669034], [-20917036], [-11264892], [444290], [34261264], [-48221134], [-44869252], [-5389066], [98167444], [-44061120], [67108722], [4349292], [-12023810], [59639888], [-98103614], [-109316918], [-36431736], [13863720], [135934840], [28220268], [-53121522], [154588854], [-186143400], [262107082], [-275827148], [1068562], [-79292416], [210169942], [-265509556], [-623529434], [130610982], [-22087352], [200187666], [-479736040], [-519734352], [969758714], [756875470], [-179867444], [-1380918694], [766252852], [-268852410], [-395002446], [-42576776], [1852945144], [-48360242], [-226078196], [-1351980232], [-1091072970], [-697867610], [1746888362], [-1303039902], [2871392396], [1159457470], [-1667032440], [634094242], [-1223680940], [-1232131460], [-3075112894], [-2447302302], [952418586], [-605200088], [-1377079164], [-4002222424], [-2886766744], [119742868], [-4789499092], [3830466352], [-2343850986], [5770848738], [-349411336], [-5100247922], [4962174176], [1425899914], [5969285380], [-3516160050], [644705624], [-2633218398], [6763465516], [-4227181138], [4002989968], [6675539882], [-5306337100], [-8658018802], [-7388913534], [9991414268], [-3298341704], [-9769712718], [-4184437036], [-696084832], [6200457622], [-11140389116], [4560959326], [5828793506], [4996239450], [12137114660], [-9234031804], [4745295194], [13454665838], [2639762976], [-3527066760], [-10382828322], [2054175968], [4874760088], [11580890272], [3467349902], [-11402290050], [23071531826], [21509122254], [-4466783840], [24389878670], [9888574638], [-11520430112], [2635163050], [-12703938672], [-11133850924], [27985245766], [2711699440], [19717533662], [18941126990], [-6776370036], [28063517256], [-10155919386], [-27802292470], [21519882076], [-31337571688], [-16245894900], [-3157263120], [2536554388], [-21028232886], [31289273354], [7917062466], [8558492412], [-8496605906], [-14798792516], [-28615714996], [33799102794], [1031335200], [-56313938960], [25553118110]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_33_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_33_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_33_8_a_a(:prec:=1)
    chi := MakeCharacter_33_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_33_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_33_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_33_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![-10, 1]>],Snew);
    return Vf;
end function;