
// Make newform 2.8.a.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2_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_2_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_2_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_2_a();"
function MakeCharacter_2_a()
    N := 2;
    order := 1;
    char_gens := [];
    v := [];
    // 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_2_a_Hecke(Kf)
    return MakeCharacter_2_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 := [[-8], [12], [-210], [1016], [1092], [1382], [14706], [-39940], [68712], [-102570], [227552], [160526], [10842], [-630748], [472656], [-1494018], [2640660], [827702], [-126004], [-1414728], [980282], [-3566800], [5672892], [-11951190], [8682146], [-10079538], [3747992], [-17985564], [12257030], [16594962], [1168256], [-7923828], [-315654], [39203780], [-21886050], [-29415448], [60554966], [57085292], [-87726504], [8569542], [18804060], [-59962498], [93986112], [-35194558], [102985326], [83637560], [-97401028], [-14645728], [-184540884], [-8754610], [-119556198], [396209040], [-256606318], [-73477548], [-202701054], [154254072], [-624018330], [-387983248], [453952286], [333769962], [537694772], [335600142], [215028956], [792061512], [-118457398], [-50730954], [273756932], [-91851214], [-1367004684], [1131432470], [-44839518], [398280600], [1634717456], [-1546331458], [-1056882220], [224909952], [1017882030], [-1475652634], [274912242], [-1634272150], [-1112798100], [922528142], [-981507888], [2849979602], [-1056218680], [1823254932], [1848459330], [-2980660294], [-2527807578], [-890290768], [2656667196], [1300933920], [-1074466024], [-783344028], [-623187940], [-2709270648], [3499434870], [-1376833158], [-2861538748], [5344671062], [-3371346964], [-5611058874], [6696895452], [1968501450], [1029257492], [3311792066], [-559410684], [-3024589518], [-5632460760], [340791722], [3854196896], [9222445262], [6536112666], [1365585380], [1540786952], [-4540181118], [11405383052], [-12639339144], [-10500443418], [9648179580], [-6582986338], [-8546489758], [8713048206], [14610865572], [-14734767268], [1315018422], [664028270], [4950344880], [8811008936], [-149413738], [-4708057300], [1696762392], [10664999312], [6228758846], [-8383341558], [-11864884990], [5566800942], [134611196], [-7415475114], [-14154174630], [-26370790108], [8062640382], [-23420185528], [5557221396], [28425575990], [10403565240], [-18058024738], [-6340342614], [12148877300], [-28711114848], [24602117126], [-12537766638], [19309723292], [32026812936], [2337034436], [22034303472], [-14327739160], [13127971170], [-38762629414], [20627937222], [-21170525604], [21487568202], [39262517816], [-56264662908], [-8434369614], [-22422959448], [34672761632], [-29647405234]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2_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_2_8_a_a(:prec:=1)
    chi := MakeCharacter_2_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_2_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_2_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;