
// Make newform 1728.4.a.r in Magma, downloaded from the LMFDB on 24 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_4_a_r();".
// 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_4_a_r();".
// 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 := 4;
    raw_aps := [[0], [0], [0], [37], [0], [19], [0], [-163], [0], [0], [-308], [-323], [0], [-520], [0], [0], [0], [-719], [-127], [0], [-919], [1387], [0], [0], [-523], [0], [1801], [0], [646], [0], [-380], [0], [0], [467], [0], [-1961], [3850], [3779], [0], [0], [0], [-1241], [0], [-3961], [0], [-863], [-3961], [3220], [0], [-4466], [0], [0], [2609], [0], [0], [0], [0], [-7289], [4030], [0], [5600], [0], [10640], [0], [-901], [0], [9899], [-7363], [0], [-10547], [0], [0], [13753], [-251], [-6103], [0], [0], [-1190], [0], [-16543], [0], [10459], [0], [-2590], [-14924], [0], [0], [12710], [0], [-7811], [0], [0], [14797], [0], [-15136], [0], [0], [0], [23921], [-20789], [-22933], [0], [0], [0], [629], [-9613], [0], [0], [0], [-29302], [-22283], [30241], [0], [26693], [28351], [0], [13160], [0], [0], [0], [13249], [-33949], [0], [0], [-16072], [0], [27523], [0], [10780], [-15050], [31376], [0], [-41021], [-17333], [0], [39023], [0], [-28747], [0], [0], [39368], [0], [33391], [0], [47143], [0], [-38411], [0], [18503], [0], [13357], [0], [-31429], [0], [-49447], [0], [-2756], [0], [40283], [0], [0], [0], [3907], [0], [0], [0], [14041], [-28910], [-59347], [0], [0], [40138], [0], [63629], [-44207], [0], [-60352], [0], [47899], [-39494], [-54557], [0], [-61490], [0], [0], [0], [-13553], [32471], [21797], [0], [52649], [0], [-42769], [0], [0], [0], [-68221], [-82529], [0], [0], [0], [-19241], [-63593], [-38323], [0], [0], [14527], [0], [0], [-51319], [-80137], [0], [94051], [0], [0], [-88489], [-15860], [0], [0], [0], [24751], [-29036], [0], [77380], [0], [-54503], [0], [0], [-41543], [0], [-99989], [-69496], [103588], [0], [-63649], [0], [-76267], [0], [0], [0], [0], [-115939], [-9161], [-121283], [0], [0], [-39797], [0], [-118816], [0], [114550], [0], [0], [-126667], [0], [0], [-116801], [121520], [0], [4970], [103069], [0], [131563], [-97631], [0], [64064], [0], [0], [142760], [0], [30529], [-144073], [51590], [120763], [-71890], [132841], [0], [111274], [5402], [0], [0], [59092], [0], [44749], [-3313], [0], [161729], [0], [-128483], [0], [0], [0], [0], [0], [121210], [0], [152101], [0], [0], [-113923], [-177301], [0], [-171236], [0], [161279], [166430], [0], [151867], [0], [182629], [0], [0], [0], [-53101], [0], [-183274], [0], [0], [-31570], [0], [168929], [-125677], [0], [-187220], [0], [182789], [-106216], [-52360], [0], [0], [-141821], [0], [64783], [0], [103952], [0], [-183707], [0], [47558], [129220], [172621], [0], [0], [-132749], [0], [0], [114478], [188480], [0], [0], [164141], [156113], [0], [-178220], [-35027], [0], [0], [0], [0], [0], [-204623], [0], [0], [0], [240623], [90791], [0], [-102059], [244511], [0], [-185113], [0], [0], [140149], [162163], [0], [0], [78839], [0], [-209827], [0], [0], [51463], [0], [99773], [0], [267388], [256303], [-145909], [0], [-261826], [0], [0], [-164467], [0], [233801], [271567], [0], [272792], [0], [112714], [0], [-191900], [0], [0], [-8372], [-274553], [0], [222731], [0], [108290], [0], [0], [-284941], [40697], [0], [0], [-249713], [0], [0], [0], [263410], [0], [0], [277130], [0], [0], [0], [-323512], [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_4_a_r();".
// 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_4_a_r(:prec:=1)
    chi := MakeCharacter_1728_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_a_r();".
// 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_4_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-37, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;