
// Make newform 1344.3.f.c in Magma, downloaded from the LMFDB on 28 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_f_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_1344_3_f_c();".
// 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_1344_3_f_c();".
// 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;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1344_f();"
function MakeCharacter_1344_f()
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    v := [2, 2, 2, 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_1344_f_Hecke();"
function MakeCharacter_1344_f_Hecke(Kf)
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    char_values := [[1, 0], [1, 0], [1, 0], [-1, 0]];
    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, 0], [0, 1], [0, -4], [1, -4], [-10, 0], [0, 4], [0, 0], [0, -12], [-14, 0], [38, 0], [0, -16], [-26, 0], [0, 40], [-26, 0], [0, -16], [-10, 0], [0, 44], [0, 20], [-74, 0], [-62, 0], [0, 24], [-46, 0], [0, 52], [0, 24], [0, -32], [0, -68], [0, 56], [-10, 0], [-74, 0], [178, 0], [-190, 0], [0, -28], [-206, 0], [0, -68], [-106, 0], [-46, 0], [0, -140], [-170, 0], [0, -56], [0, 100], [38, 0], [0, 108], [82, 0], [50, 0], [278, 0], [0, -168], [-74, 0], [0, -176], [0, 4], [0, -164], [178, 0], [34, 0], [0, 112], [0, 108], [0, -80], [-62, 0], [0, 132], [0, 192], [-26, 0], [-494, 0], [0, -116], [0, 284], [0, -12], [0, -8], [0, -152], [-250, 0], [646, 0], [146, 0], [-106, 0], [0, -364], [0, 176], [-494, 0], [0, -320], [-218, 0], [550, 0], [0, -32], [-394, 0], [0, -60], [178, 0], [0, -376], [0, -412], [-602, 0], [34, 0], [0, 240], [0, -56], [470, 0], [514, 0], [-478, 0], [0, 388], [-430, 0], [0, -492], [0, 224], [818, 0], [-490, 0], [-554, 0], [0, -168], [0, -556], [0, 72], [0, -36], [118, 0], [406, 0], [998, 0], [0, 212], [-62, 0], [-410, 0], [0, -480], [0, 316], [0, -576], [1186, 0], [0, 232], [0, 80], [-26, 0], [-734, 0], [0, -484], [-286, 0], [658, 0], [0, -188], [0, 264], [614, 0], [-442, 0], [0, -20], [386, 0], [0, -452], [-298, 0], [0, 500], [-154, 0], [-890, 0], [0, -48], [0, 136], [0, -44], [982, 0], [-686, 0], [290, 0], [358, 0], [0, 376], [0, -448], [0, -132], [0, -44], [0, 500], [802, 0], [0, -516], [-1450, 0], [-1246, 0], [-586, 0], [0, 468], [0, 8], [0, 140], [0, -808], [0, 172], [-1070, 0], [502, 0], [0, 576], [118, 0], [0, -296], [-1466, 0], [-542, 0], [-1630, 0], [0, -112], [0, 696], [0, -508], [-1402, 0], [-110, 0], [1202, 0], [0, 156], [-1310, 0], [0, -904], [818, 0], [0, -100], [530, 0], [0, -244], [758, 0], [0, -44], [-446, 0], [-46, 0], [0, -352], [0, -776], [1222, 0], [1142, 0], [0, 1080], [0, -988], [386, 0], [0, -476], [358, 0], [0, -1176], [-542, 0], [0, 172], [118, 0], [0, 1124], [146, 0], [0, 864], [0, 672], [-1450, 0], [790, 0], [0, -716], [1478, 0], [0, -536], [-1582, 0], [-1034, 0], [0, -496], [0, 648], [-1498, 0], [0, -512], [0, -788], [0, 1120], [0, 1020], [0, -204], [0, -720], [-826, 0], [1378, 0], [0, -452], [-2350, 0], [0, -628], [-430, 0], [0, -20], [0, -632], [0, -1064], [-430, 0], [0, 608], [-350, 0], [-250, 0], [358, 0], [0, 8], [130, 0], [-1054, 0], [0, -1228], [646, 0], [0, -712], [2050, 0], [0, -584], [1718, 0], [-1994, 0], [0, 1236], [1778, 0], [-398, 0], [0, 1148], [0, -208], [0, 560], [-394, 0], [1334, 0], [0, -696], [-538, 0], [0, -372], [0, 1208], [-458, 0], [0, -512], [0, 152], [0, -1296], [0, 932], [2662, 0], [178, 0], [-2954, 0], [0, -1296], [706, 0], [0, -8], [0, -860], [1190, 0], [-602, 0], [0, 524], [0, 444], [0, 840], [-190, 0], [-154, 0], [0, -932], [0, -300], [0, 112], [0, -796], [-442, 0], [0, 440], [1318, 0], [1046, 0], [0, 1060], [-1322, 0], [0, -1304], [1154, 0], [0, -208], [0, 264], [-2410, 0], [2710, 0], [1250, 0], [0, 1780], [0, 448], [-574, 0], [0, 1784], [0, -4], [0, -548], [2530, 0], [-286, 0], [-1546, 0], [0, 136], [0, 1104], [-442, 0], [0, 372], [-1982, 0], [0, 1052], [694, 0], [0, 788], [0, -176], [0, -1652], [0, 492], [0, 164], [0, -1000], [-1978, 0], [1106, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_3_f_c();".
// 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_1344_3_f_c(:prec:=2)
    chi := MakeCharacter_1344_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1344_3_f_c();".
// 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_1344_3_f_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![48, 0, 1]>,<11,R![10, 1]>],Snew);
    return Vf;
end function;