
// Make newform 6005.2.a.b in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6005_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_6005_2_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_6005_2_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_6005_a();"
function MakeCharacter_6005_a()
    N := 6005;
    order := 1;
    char_gens := [1202, 11];
    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_6005_a_Hecke(Kf)
    return MakeCharacter_6005_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 := 2;
    raw_aps := [[-1], [0], [1], [4], [-4], [4], [-4], [-4], [-6], [2], [-4], [4], [-2], [0], [6], [-8], [12], [14], [-4], [0], [0], [16], [6], [6], [12], [-10], [8], [-6], [-10], [-12], [10], [-4], [22], [12], [-6], [0], [-8], [8], [16], [-14], [16], [22], [16], [-14], [18], [-8], [4], [4], [10], [-6], [-18], [-24], [14], [-12], [-2], [-12], [10], [-20], [24], [-18], [-14], [4], [16], [8], [28], [-14], [24], [-26], [12], [26], [4], [24], [28], [12], [-12], [-6], [-10], [-18], [30], [-14], [28], [2], [20], [34], [20], [-20], [18], [-12], [6], [6], [12], [24], [-26], [12], [16], [-2], [18], [10], [26], [-2], [-22], [-12], [4], [42], [-44], [-28], [-28], [-30], [-24], [-26], [42], [-10], [12], [-32], [16], [34], [-14], [18], [-22], [-36], [-2], [-20], [32], [2], [-36], [34], [-22], [40], [52], [4], [0], [-48], [40], [20], [10], [-14], [28], [14], [12], [18], [-16], [42], [40], [0], [-50], [4], [36], [14], [-4], [-20], [-6], [-18], [-14], [30], [32], [-24], [0], [18], [0], [-42], [-34], [-32], [38], [0], [-38], [-2], [24], [-8], [58], [-16], [-20], [-30], [-48], [-46], [-16], [10], [60], [38], [-30], [14], [48], [-56], [-38], [-32], [-40], [-6], [-18], [58], [-10], [0], [-10], [36], [-36], [-30], [8], [-54], [-1], [-26], [-6], [-26], [18], [-16], [14], [10], [28], [70], [32], [-26], [62], [-28], [38], [26], [26], [38], [56], [26], [64], [2], [-42], [-14], [-38], [68], [6], [46], [30], [34], [-40], [-60], [-74], [-68], [62], [-20], [-24], [22], [-42], [-48], [-50], [24], [-68], [40], [54], [68], [52], [-14], [24], [0], [0], [40], [52], [-50], [-16], [18], [-74], [-46], [-42], [-36], [2], [38], [-76], [34], [-58], [-44], [26], [76], [-32], [-36], [-78], [-46], [-58], [66], [46], [-6], [40], [28], [-54], [14], [70], [-26], [-54], [-60], [12], [40], [66], [30], [42], [48], [-38], [2], [-8], [66], [-42], [46], [-18], [-24], [18], [-18], [-80], [82], [0], [42], [12], [-18], [-80], [-24], [84], [-48], [12], [26], [16], [-36], [6], [-30], [-30], [-52], [-12], [6], [88], [-16], [-18], [-30], [64], [32], [66], [38], [4], [62], [28], [56], [26], [-66], [50], [-24], [32], [-54], [-12], [-28], [-18], [26], [-22], [-34], [80], [62], [30], [64], [-72], [-52], [90], [38], [80], [-62], [-36], [2], [-46], [28], [26], [6], [0], [60], [-58], [32], [62], [34], [60], [-44], [30], [-48], [30], [34], [-82], [-12], [20], [6], [-90], [40], [-40], [60], [-72], [52], [26], [30], [-42], [96], [88], [36], [-20], [46], [-88], [-92], [102], [28], [-46], [-28], [0], [-52], [-80], [-66], [0], [-66], [-96], [18], [90], [92], [66], [-46], [6], [20], [58], [2], [-8], [12], [16], [-32], [24], [-96], [-56], [-42], [72], [86], [24], [56], [30], [-14], [72], [64], [-92], [56], [-72], [-42], [-92], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6005_2_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_6005_2_a_b(:prec:=1)
    chi := MakeCharacter_6005_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6005_2_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_6005_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>],Snew);
    return Vf;
end function;