
// Make newform 1728.2.a.h in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_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_1728_2_a_h();".
// 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_1728_2_a_h();".
// 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_1728_a();"
function MakeCharacter_1728_a()
    N := 1728;
    order := 1;
    char_gens := [703, 325, 1217];
    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_1728_a_Hecke(Kf)
    return MakeCharacter_1728_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], [0], [-2], [3], [-6], [3], [-2], [3], [6], [8], [0], [-7], [8], [12], [6], [-4], [-6], [1], [3], [12], [-15], [9], [12], [-10], [9], [16], [15], [6], [-6], [14], [-12], [0], [-14], [21], [-4], [-3], [14], [-3], [-18], [-8], [12], [-9], [-6], [-5], [-2], [15], [-15], [-24], [-12], [6], [-8], [12], [-3], [0], [-20], [-12], [10], [-15], [18], [8], [-12], [2], [-12], [18], [-1], [4], [-3], [9], [-12], [1], [20], [18], [3], [-23], [27], [0], [-34], [-2], [-20], [21], [-30], [15], [30], [-2], [0], [12], [-22], [-30], [2], [15], [-6], [12], [15], [-42], [-24], [-42], [-10], [-14], [-9], [15], [9], [2], [24], [46], [3], [11], [-18], [28], [24], [-26], [3], [-11], [2], [-9], [-27], [4], [36], [24], [28], [12], [5], [-1], [-2], [24], [-12], [-46], [51], [-36], [24], [18], [12], [6], [45], [27], [-34], [-49], [20], [27], [40], [44], [-36], [-50], [9], [-6], [-45], [-36], [53], [-4], [-27], [-42], [13], [-14], [9], [-36], [39], [48], [24], [28], [7], [-22], [-18], [46], [33], [6], [44], [-6], [-21], [-26], [-3], [10], [-36], [-58], [6], [21], [27], [44], [-36], [26], [-3], [-46], [57], [-12], [-6], [-22], [-24], [4], [15], [9], [9], [30], [21], [-36], [21], [26], [-18], [10], [-13], [27], [-10], [-36], [-38], [57], [-45], [33], [-24], [4], [-27], [42], [-38], [-45], [47], [-50], [-15], [18], [72], [-53], [48], [62], [36], [-20], [-57], [24], [40], [48], [-18], [-55], [-2], [30], [-69], [-6], [3], [0], [-36], [34], [-15], [-54], [-51], [-22], [12], [-60], [-36], [-69], [-51], [-7], [-26], [-6], [-51], [-24], [-48], [48], [42], [34], [30], [-15], [-26], [54], [39], [-12], [-64], [-2], [27], [-42], [-21], [-79], [-28], [-36], [34], [4], [24], [2], [49], [33], [54], [21], [-6], [-57], [-54], [46], [-34], [48], [-66], [-36], [24], [1], [-27], [-30], [-51], [44], [-9], [-70], [70], [48], [-4], [72], [-46], [-40], [-81], [-26], [-24], [39], [-5], [32], [0], [-48], [21], [-54], [78], [-9], [-24], [-47], [-66], [40], [8], [-39], [-6], [-54], [18], [30], [66], [10], [63], [67], [-28], [36], [-20], [-3], [12], [0], [-78], [-86], [-81], [-32], [33], [-24], [48], [-36], [-43], [-86], [18], [60], [17], [-70], [4], [57], [-40], [-90], [6], [12], [12], [62], [-57], [33], [88], [24], [77], [34], [-12], [-48], [-80], [12], [53], [32], [12], [-78], [45], [71], [2], [-33], [47], [-42], [45], [-72], [32], [-9], [-57], [-30], [-24], [-29], [4], [33], [-50], [-40], [81], [-44], [39], [84], [-84], [75], [45], [-6], [30], [-32], [18], [63], [-42], [-51], [-51], [26], [12], [-32], [22], [-40], [24], [32], [20], [-24], [-57], [-44], [-75], [54], [-82], [-28], [-6], [93], [-39], [2], [-18], [33], [8], [-36], [-34], [-34], [78], [96], [-74], [32], [42], [-20], [-84], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_2_a_h();".
// 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_1728_2_a_h(:prec:=1)
    chi := MakeCharacter_1728_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_1728_2_a_h();".
// 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_1728_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-3, 1]>,<11,R![6, 1]>],Snew);
    return Vf;
end function;