
// Make newform 441.3.n.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_n_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_n_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_3_n_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_n();"
function MakeCharacter_441_n()
    N := 441;
    order := 6;
    char_gens := [344, 199];
    v := [1, 2];
    // 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_n_Hecke();"
function MakeCharacter_441_n_Hecke(Kf)
    N := 441;
    order := 6;
    char_gens := [344, 199];
    char_values := [[0, 1], [-1, 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 := [[2, -1], [-3, 0], [2, -4], [0, 0], [-1, 2], [0, -4], [18, -9], [11, -11], [16, -32], [26, 26], [32, -32], [34, -34], [14, -7], [61, -61], [-56, 28], [0, 0], [29, 29], [0, 56], [31, -31], [-18, 36], [0, 65], [0, -38], [28, 28], [-72, -72], [-115, 115], [-26, 52], [40, 0], [-81, -81], [0, 52], [-104, 52], [-16, 0], [92, -184], [-109, 218], [0, 5], [88, -176], [20, 0], [-40, 40], [106, -106], [-220, 110], [268, -134], [-72, 36], [232, 0], [-268, 134], [265, -265], [72, -144], [0, 290], [0, 94], [-52, 52], [109, -218], [-266, 0], [117, 117], [-464, 232], [-119, 0], [225, -450], [101, -202], [26, -52], [-216, 108], [-268, 268], [56, 0], [-28, -28], [374, -374], [-292, 146], [-533, 0], [-142, -142], [0, 155], [56, -28], [0, -2], [0, -77], [-65, -65], [416, -416], [1, -2], [342, 342], [358, 0], [-580, 0], [83, 0], [-322, 644], [-298, 596], [362, -362], [-227, 454], [221, -221], [-904, 452], [682, -682], [324, -162], [295, 0], [0, 812], [-106, 53], [-369, 738], [0, -65], [460, 460], [0, -734], [117, 117], [-350, 700], [0, 106], [-266, 133], [-787, 0], [-360, 720], [-124, 248], [-234, 117], [-250, 250], [-650, 650], [-623, 623], [-612, 306], [65, 65], [-754, 377], [-545, 545], [0, -871], [1, 1], [-108, -108], [-326, -326], [461, -461], [112, 0], [0, -902], [-410, 205], [799, 0], [830, 0], [-217, 434], [0, -13], [540, -270], [-218, 436], [-568, -568], [-382, 382], [-578, 578], [808, -404], [1206, -603], [0, 182], [0, 0], [0, 700], [342, 342], [-664, 664], [670, 0], [0, -317], [-716, 358], [1310, 0], [218, 0], [380, -760], [0, 1022], [1368, -684], [-130, 130], [-364, 182], [162, -81], [-299, 0], [-700, 350], [814, -814], [-828, 1656], [0, -718], [460, 460], [1424, -1424], [-404, 808], [-311, 0], [-594, -594], [104, 0], [-36, 72], [119, 0], [686, -1372], [695, 0], [-1000, -1000], [-56, 56], [-920, 460], [-470, 0], [-232, -232], [2, -1], [477, -954], [0, -1202], [108, 108], [241, 241], [-674, 1348], [0, 1420], [-524, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_3_n_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_3_n_a(:prec:=2)
    chi := MakeCharacter_441_n();
    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_n_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_3_n_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![3, -3, 1]>,<13,R![16, 4, 1]>],Snew);
    return Vf;
end function;