
// Make newform 441.4.e.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_441_4_e_a();".
// 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_441_4_e_a();".
// 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
        poly := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_441_e();"
function MakeCharacter_441_e()
    N := 441;
    order := 3;
    char_gens := [344, 199];
    v := [3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_e_Hecke();"
function MakeCharacter_441_e_Hecke(Kf)
    N := 441;
    order := 3;
    char_gens := [344, 199];
    char_values := [[1, 0], [0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, -5], [0, 0], [0, 0], [0, 0], [-68, 68], [0, 0], [0, 0], [0, 0], [0, -40], [166, 0], [0, 0], [0, -450], [0, 0], [-180, 0], [0, 0], [590, -590], [0, 0], [0, 0], [740, -740], [-688, 0], [0, 0], [0, 1384], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -1580], [54, -54], [670, 0], [-2000, 0], [0, 0], [3110, -3110], [0, 0], [0, 814], [2952, -2952], [0, 0], [0, -1780], [0, 0], [0, 0], [-2084, 2084], [0, 0], [0, -4072], [4590, -4590], [2210, 0], [0, 0], [5868, 0], [0, 0], [0, 0], [0, 0], [0, -4730], [7376, 0], [0, 0], [0, 0], [0, 0], [7520, -7520], [0, 0], [0, 0], [-7310, 7310], [-4342, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -6970], [0, -10908], [-3330, 0], [-4100, 4100], [0, 0], [0, 0], [0, -8104], [0, 0], [0, 13970], [11916, 0], [0, 0], [-10526, 10526], [0, 0], [0, 1598], [0, 0], [0, 0], [15262, 0], [-8608, 8608], [0, 0], [0, 0], [0, 18580], [2686, 0], [0, -8010], [0, 0], [-8440, 0], [0, 0], [0, 0], [-21240, 21240], [-20372, 0], [0, 7236], [0, 0], [0, 0], [0, 0], [0, 0], [0, -15878], [12980, 0], [20470, -20470], [0, 0], [0, -26906], [6788, -6788], [0, 0], [0, 0], [0, 0], [-24736, 24736], [0, 0], [0, 0], [-15010, 15010], [-30550, 0], [0, 0], [-26192, 0], [8878, -8878], [0, 0], [0, 0], [0, 27050], [1804, 0], [0, 0], [-33570, 0], [0, 0], [-34060, 34060], [0, 0], [4198, 0], [0, -12546], [0, 0], [0, 0], [0, 0], [25324, -25324], [-25160, 0], [0, 2448], [-34830, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [37354, -37354], [0, 0], [0, -43538], [46240, -46240], [23980, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -20200], [0, 6550], [0, 0], [30060, 0], [0, 0], [-52740, 52740], [39632, 0], [0, -21744], [0, 0], [0, 0], [0, 0], [0, 48820], [-29290, 0], [52040, 0], [0, 0], [-37490, 37490], [0, 0], [-57528, 57528], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_4_e_a();".
// 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_441_4_e_a(:prec:=2)
    chi := MakeCharacter_441_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_441_4_e_a();".
// 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_441_4_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![25, 5, 1]>,<5,R![0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;