
// Make newform 24.6.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_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_24_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_24_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_24_a();"
function MakeCharacter_24_a()
    N := 24;
    order := 1;
    char_gens := [7, 13, 17];
    v := [1, 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_24_a_Hecke(Kf)
    return MakeCharacter_24_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 := [[0], [-9], [94], [144], [-380], [814], [-862], [-1156], [-488], [-5466], [9560], [-10506], [-5190], [-17084], [3168], [-24770], [17380], [4366], [-5284], [8360], [39466], [42376], [-61828], [-63078], [-16318], [134766], [45184], [143124], [153070], [-64494], [33256], [-149284], [148602], [-324012], [-59986], [91520], [345550], [-634164], [-415848], [-324282], [606828], [463254], [955008], [-256126], [111150], [-516800], [400972], [-206168], [363468], [-589274], [730090], [472496], [993042], [1936532], [735746], [-542984], [438614], [644408], [-1419530], [645242], [1936964], [-1654290], [-2782356], [1005096], [535386], [2962662], [1559348], [3615570], [-1764876], [553134], [-1174078], [-1189752], [-1353272], [2793062], [-3373036], [4727808], [188254], [-524578], [-3086526], [-4601958], [2094924], [987206], [3259328], [7314610], [-7314192], [-7074028], [7480786], [2971082], [-8290778], [955064], [-8190180], [-3397744], [1948144], [-3925740], [2071020], [-9165000], [6915174], [-2260774], [8255236], [1904670], [1570604], [2368486], [6649756], [-8945654], [-4195260], [-6248446], [-4552844], [210802], [3059448], [7370938], [2122568], [-10968874], [-7289526], [-2236748], [-7061600], [18018898], [1072524], [20305448], [19729158], [12951084], [20188838], [-18398942], [-4727906], [-13808444], [12896444], [-8810778], [18336902], [22623808], [-12565824], [833534], [8841884], [-26213960], [6171432], [-6707338], [-10811350], [-13304830], [-12905298], [6777052], [14955094], [-8658406], [-19832252], [30222750], [-26469184], [8818820], [-23105282], [26508360], [-7842010], [-22656342], [-26739084], [-12995744], [-3168450], [38581106], [-23122372], [-26260216], [31321348], [4498768], [-3038240], [-193966], [-32321334], [-45028794], [10449516], [43134282], [-14398240], [3002724], [-37516350], [-2572440], [-54188744], [35267350]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_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_24_6_a_b(:prec:=1)
    chi := MakeCharacter_24_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_24_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_24_6_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![-94, 1]>],Snew);
    return Vf;
end function;