
// Make newform 1444.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1444_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_1444_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_1444_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_1444_a();"
function MakeCharacter_1444_a()
    N := 1444;
    order := 1;
    char_gens := [723, 1085];
    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_1444_a_Hecke(Kf)
    return MakeCharacter_1444_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], [0], [-4], [-1], [3], [0], [5], [7], [4], [10], [-5], [-5], [-7], [11], [3], [11], [-3], [11], [15], [-13], [0], [3], [-5], [-1], [0], [20], [3], [14], [-3], [15], [-5], [9], [3], [-16], [7], [-4], [-15], [15], [-12], [-5], [16], [15], [2], [-7], [-9], [-25], [-20], [2], [-21], [12], [19], [-31], [23], [-9], [27], [31], [10], [7], [-9], [-30], [27], [-20], [11], [15], [20], [-5], [-5], [10], [30], [15], [25], [-10], [-12], [-29], [3], [-25], [19], [-17], [12], [-1], [21], [-25], [-13], [25], [22], [-22], [11], [-20], [-20], [-23], [-8], [-1], [-5], [21], [15], [26], [-19], [-1], [25], [-5], [36], [22], [-4], [-10], [15], [-5], [45], [-26], [32], [-29], [-45], [-20], [41], [39], [19], [32], [-30], [5], [31], [-10], [10], [16], [36], [35], [3], [7], [7], [-30], [-13], [-25], [29], [-17], [30], [7], [-21], [52], [18], [-18], [21], [-9], [-31], [-7], [30], [-25], [39], [15], [13], [-36], [-5], [6], [25], [25], [35], [-24], [-40], [23], [23], [19], [27], [-25], [-45], [35], [-10], [-45], [-5], [-53], [-53], [51], [-12], [35], [39], [14], [45], [15], [-19], [-45], [40], [35], [-23], [4], [-41], [-5], [-24], [-42], [-25], [31], [-10], [24], [19], [-55], [4], [51], [25], [-9], [-25], [15], [-10], [-24], [-21], [1], [-5], [-61], [-25], [-37], [-7], [-51], [-57], [-40], [35], [-49], [60], [-3], [24], [-45], [-33], [30], [-32], [-45], [39], [-40], [23], [-25], [-13], [-45], [-10], [-9], [35], [36], [39], [17], [-24], [-50], [-64], [63], [-18], [50], [-13], [9], [31], [-20], [-35], [11], [-9], [0], [37], [53], [-37], [-75], [-38], [71], [-40], [-25], [-25], [-23], [-9], [20], [-33], [7], [1], [15], [63], [-21], [27], [28], [-18], [-6], [-55], [75], [42], [28], [-45], [32], [23], [75], [12], [-65], [-5], [69], [60], [4], [57], [-30], [7], [-75], [30], [27], [67], [38], [-66], [-72], [-45], [44], [-1], [-22], [-55], [35], [-29], [12], [-5], [-73], [37], [-4], [-69], [23], [45], [-13], [45], [74], [-44], [-33], [31], [36], [15], [78], [-3], [-89], [35], [-50], [-5], [-45], [-9], [-19], [-45], [39], [25], [-20], [-65], [-49], [31], [3], [-37], [4], [-75], [-43], [-30], [-30], [86], [-32], [-29], [15], [-33], [-40], [11], [-53], [-21], [57], [11], [18], [-11], [83], [-49], [24], [23], [46], [-61], [-81], [-5], [-75], [-17], [-49], [-75], [80], [-25], [-25], [78], [39], [79], [57], [12], [-1], [43], [35], [50], [-97], [56], [95], [-33], [-33], [66], [14], [-47], [15], [-72], [-25], [16], [-5], [-11], [-20], [95], [-1], [-12], [-35], [21], [-25], [67], [-82], [55], [-9], [95], [-29], [72], [-5], [-9], [-25], [-65], [-34], [-35], [-40], [-101], [15], [16], [-64], [58], [66], [-87], [-40], [27], [-35], [-33], [-73], [80], [13], [-74], [-18], [-76], [43], [-76], [-13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1444_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1444_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_1444_2_a_b(:prec:=1)
    chi := MakeCharacter_1444_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_1444_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_1444_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1444_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>],Snew);
    return Vf;
end function;