
// Make newform 441.3.m.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_m_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_3_m_d();".
// 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_3_m_d();".
// 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_m();"
function MakeCharacter_441_m()
    N := 441;
    order := 6;
    char_gens := [344, 199];
    v := [6, 5];
    // 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_m_Hecke();"
function MakeCharacter_441_m_Hecke(Kf)
    N := 441;
    order := 6;
    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 := 3;
    raw_aps := [[0, 1], [0, 0], [-8, 4], [0, 0], [10, -10], [4, -8], [0, 0], [24, -12], [0, -14], [38, 0], [16, 16], [0, -26], [40, -80], [26, 0], [32, -16], [10, -10], [-44, -44], [-40, 20], [-74, 74], [62, 0], [-24, -24], [0, 46], [-52, 104], [-48, 24], [32, -64], [68, 68], [112, -56], [0, 10], [-74, 74], [-178, 0], [-190, 0], [-56, 28], [-206, 206], [-68, 136], [0, 106], [46, -46], [-140, -140], [0, -170], [-56, 112], [200, -100], [-38, 38], [108, -216], [0, 82], [-50, 50], [278, 0], [168, 168], [74, 0], [176, -352], [-4, -4], [328, -164], [0, 178], [-34, 0], [-112, -112], [-108, 216], [160, -80], [-62, 62], [-132, -132], [384, -192], [-26, 26], [494, 0], [-116, -116], [-284, 568], [-12, 24], [-8, -8], [-304, 152], [0, 250], [0, 646], [146, 0], [106, -106], [-364, 728], [176, 176], [0, -494], [320, 320], [0, -218], [-550, 0], [64, -32], [394, -394], [120, -60], [0, 178], [376, 376], [412, -824], [602, 0], [34, -34], [-240, 480], [-112, 56], [0, -470], [-514, 0], [0, 478], [-388, 776], [-430, 0], [-984, 492], [224, 224], [-818, 818], [-490, 0], [0, -554], [-168, 336], [-1112, 556], [72, 72], [72, -36], [0, 118], [-406, 0], [-998, 998], [-212, -212], [0, -62], [-410, 410], [480, 480], [-316, 632], [1152, -576], [1186, -1186], [-232, 464], [160, -80], [-26, 26], [734, 0], [-484, -484], [-286, 0], [658, -658], [-188, 376], [264, 264], [0, -614], [-442, 0], [-20, -20], [386, 0], [-904, 452], [298, -298], [-1000, 500], [-154, 0], [0, -890], [96, -48], [-136, 272], [88, -44], [982, -982], [686, 0], [0, -290], [-358, 0], [-752, 376], [448, -896], [132, 132], [-44, -44], [-500, 1000], [802, -802], [-516, 1032], [0, 1450], [1246, -1246], [-586, 0], [468, 468], [8, -16], [140, -280], [-808, -808], [-344, 172], [0, -1070], [0, 502], [576, -1152], [-118, 0], [592, -296], [-1466, 1466], [542, 0], [0, 1630], [224, -112], [-696, 1392], [508, 508], [0, 1402], [110, 0], [1202, 0], [312, -156], [-1310, 1310], [-904, -904], [-818, 818], [-100, -100]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_3_m_d();".
// 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_3_m_d(:prec:=2)
    chi := MakeCharacter_441_m();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_m_d();".
// 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_3_m_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![1, -1, 1]>,<5,R![48, 12, 1]>],Snew);
    return Vf;
end function;