
// Make newform 4028.2.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4028_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_4028_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_4028_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_4028_a();"
function MakeCharacter_4028_a()
    N := 4028;
    order := 1;
    char_gens := [2015, 2757, 2281];
    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_4028_a_Hecke(Kf)
    return MakeCharacter_4028_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], [2], [-4], [-3], [4], [7], [1], [-8], [-6], [9], [-4], [6], [-11], [4], [-1], [4], [-10], [-13], [-15], [0], [-8], [-4], [8], [4], [-16], [7], [-6], [-2], [-2], [-7], [-9], [-12], [-14], [18], [16], [-4], [-21], [25], [-15], [5], [5], [-20], [17], [-22], [25], [24], [18], [2], [-11], [16], [-28], [8], [-2], [25], [-10], [0], [9], [-2], [-22], [0], [6], [16], [17], [-14], [26], [4], [-21], [-3], [-8], [-18], [-4], [11], [5], [25], [-12], [10], [-10], [35], [-24], [-18], [-21], [16], [-6], [-10], [18], [6], [-12], [7], [-28], [-3], [10], [16], [30], [-30], [-26], [-42], [-2], [36], [-1], [-2], [24], [11], [-15], [-40], [37], [48], [5], [4], [14], [22], [-44], [22], [-23], [-26], [18], [7], [-15], [11], [-48], [42], [-46], [33], [-36], [38], [-20], [-34], [44], [-13], [49], [16], [20], [-30], [-38], [4], [22], [-33], [20], [-45], [6], [30], [36], [-31], [23], [-25], [54], [-20], [0], [-49], [4], [-44], [48], [-24], [24], [-8], [18], [-20], [7], [43], [38], [-28], [-2], [-33], [-40], [-30], [46], [32], [-33], [41], [-14], [12], [3], [-31], [6], [-60], [-37], [40], [6], [-11], [18], [-10], [33], [26], [52], [-9], [-11], [-4], [-22], [-20], [-48], [42], [-24], [-45], [-40], [26], [5], [40], [-27], [24], [41], [-20], [-1], [-23], [-15], [49], [60], [-59], [6], [-41], [21], [22], [-10], [6], [-4], [-56], [60], [-10], [-24], [-38], [6], [53], [23], [-9], [-74], [-46], [-6], [-27], [-15], [-28], [44], [64], [-46], [-58], [-1], [-45], [-12], [-10], [-41], [9], [55], [7], [21], [-11], [26], [44], [34], [64], [40], [-34], [-40], [-54], [19], [-51], [-18], [-18], [73], [14], [26], [-54], [-2], [-41], [8], [14], [-22], [21], [36], [-50], [58], [-1], [1], [35], [-12], [-47], [64], [-22], [-66], [21], [8], [64], [40], [7], [-72], [8], [22], [-32], [-3], [-16], [-19], [-7], [-9], [-53], [36], [27], [23], [46], [10], [52], [54], [-47], [10], [8], [-42], [40], [8], [-32], [-1], [78], [-26], [-31], [-12], [45], [50], [-33], [16], [34], [65], [-14], [60], [-23], [-66], [18], [-25], [30], [-59], [-22], [-27], [14], [-29], [53], [-25], [6], [-84], [-49], [-72], [-57], [8], [10], [-18], [-80], [52], [-29], [32], [-9], [-43], [-44], [10], [-19], [-82], [-23], [9], [41], [-50], [-63], [-88], [24], [47], [-60], [-50], [42], [-36], [-77], [-38], [-28], [-63], [17], [57], [-58], [37], [58], [12], [36], [81], [-6], [71], [-78], [60], [56], [65], [86], [22], [30], [-6], [14], [9], [44], [30], [-78], [-94], [-98], [4], [-71], [14], [4], [-8], [48], [10], [-58], [-45], [-86], [-5], [40], [-54], [52], [-37], [-6], [68], [-78], [-25], [-4], [71], [30], [-73], [-12], [-7], [-32], [-18], [6], [70], [16], [-79], [32], [-53], [12], [-12], [-6], [-82], [-105], [57], [38], [36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4028_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4028_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_4028_2_a_b(:prec:=1)
    chi := MakeCharacter_4028_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_4028_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_4028_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4028_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![-2, 1]>],Snew);
    return Vf;
end function;