
// Make newform 4029.2.a.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4029_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_4029_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_4029_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_4029_a();"
function MakeCharacter_4029_a()
    N := 4029;
    order := 1;
    char_gens := [2687, 3556, 3163];
    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_4029_a_Hecke(Kf)
    return MakeCharacter_4029_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 := [[0], [1], [1], [-2], [3], [-5], [1], [3], [-1], [2], [-4], [-8], [-11], [3], [12], [6], [12], [-4], [-8], [0], [-10], [-1], [14], [-4], [-14], [10], [-13], [-15], [-14], [-9], [5], [9], [-18], [-20], [-10], [4], [13], [-16], [7], [15], [-20], [0], [-2], [26], [11], [-22], [-8], [7], [-5], [-10], [-27], [-18], [-30], [0], [2], [-10], [27], [1], [-8], [-20], [22], [30], [12], [0], [8], [2], [9], [-30], [12], [-9], [6], [-20], [-10], [22], [16], [16], [30], [38], [-15], [-17], [-8], [-17], [-28], [19], [-30], [10], [18], [1], [20], [24], [-38], [3], [26], [32], [4], [-5], [4], [-27], [-36], [-4], [22], [6], [-6], [30], [-40], [3], [-28], [0], [38], [-16], [14], [25], [14], [-42], [-31], [29], [36], [14], [1], [20], [-5], [44], [-25], [9], [-10], [26], [40], [9], [20], [-30], [13], [48], [30], [17], [8], [27], [18], [-48], [-18], [23], [34], [-19], [-2], [23], [18], [9], [-6], [10], [-4], [-34], [-54], [-46], [49], [-55], [-4], [-17], [-25], [-25], [-2], [-46], [-52], [-36], [-19], [0], [-12], [43], [24], [12], [-34], [26], [-12], [49], [35], [-39], [41], [-39], [18], [49], [-55], [19], [-13], [-12], [-26], [56], [16], [-40], [38], [19], [38], [32], [64], [-20], [43], [60], [13], [26], [-12], [-24], [54], [-8], [63], [-12], [1], [-39], [42], [-56], [-40], [68], [-17], [-60], [-38], [30], [10], [34], [-19], [-32], [11], [24], [-37], [22], [57], [-32], [-24], [32], [-54], [-70], [-67], [21], [1], [60], [35], [26], [-64], [-68], [20], [28], [44], [33], [45], [42], [39], [-28], [-41], [-51], [-43], [-27], [-2], [7], [-65], [14], [-57], [-10], [0], [-58], [32], [46], [28], [-34], [-33], [66], [16], [-30], [18], [-24], [-29], [-29], [60], [-22], [66], [-54], [40], [-43], [73], [-80], [-34], [-1], [26], [-35], [-77], [30], [10], [-38], [-12], [-22], [-68], [-6], [4], [-54], [-45], [-54], [-31], [39], [-84], [-61], [68], [-53], [-40], [-28], [-49], [28], [49], [-60], [-30], [81], [64], [-48], [12], [30], [20], [26], [60], [-82], [-57], [69], [24], [21], [18], [-17], [-50], [6], [-30], [-4], [-26], [43], [-30], [37], [32], [-40], [36], [-9], [-12], [-5], [-38], [80], [38], [35], [-69], [75], [-6], [-41], [-27], [-18], [-7], [7], [-32], [-39], [-26], [-67], [-27], [2], [13], [-62], [64], [-56], [-45], [40], [-14], [-12], [11], [-82], [-44], [-63], [-66], [44], [-59], [55], [-23], [-64], [-38], [0], [-18], [19], [96], [-52], [-82], [21], [-32], [-9], [30], [17], [63], [28], [-52], [9], [-54], [-51], [-71], [-38], [54], [-28], [-10], [-31], [90], [3], [92], [36], [-76], [74], [40], [-70], [34], [-38], [5], [30], [44], [-56], [14], [-12], [-79], [21], [8], [66], [24], [60], [85], [66], [69], [-10], [-61], [42], [0], [-78], [69], [0], [6], [-26], [32], [-85], [95], [-87]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4029_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4029_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_4029_2_a_b(:prec:=1)
    chi := MakeCharacter_4029_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_4029_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_4029_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4029_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![-1, 1]>],Snew);
    return Vf;
end function;