
// Make newform 882.3.c.b in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_c_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_882_3_c_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_882_3_c_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
        poly := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0], [0, 4, 0, 1]];
    Rf_basisdens := [1, 2, 1, 2];
    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_882_c();"
function MakeCharacter_882_c()
    N := 882;
    order := 2;
    char_gens := [785, 199];
    v := [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_882_c_Hecke();"
function MakeCharacter_882_c_Hecke(Kf)
    N := 882;
    order := 2;
    char_gens := [785, 199];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 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, -1, 0, 0], [0, 0, 0, 0], [0, 0, 2, -2], [0, 0, 0, 0], [-6, 0, 0, 0], [0, 0, 1, 8], [0, 0, 8, -2], [0, 0, 7, -2], [12, -18, 0, 0], [0, -24, 0, 0], [0, 0, 17, 6], [-11, 12, 0, 0], [0, 0, -26, -4], [7, 6, 0, 0], [0, 0, -22, -2], [60, -18, 0, 0], [0, 0, 4, 14], [0, 0, 12, -8], [-55, -42, 0, 0], [-78, 42, 0, 0], [0, 0, -11, 40], [5, -66, 0, 0], [0, 0, 10, 38], [0, 0, -12, 0], [0, 0, 12, -4], [0, 0, -6, 48], [0, 0, -7, -38], [-144, -18, 0, 0], [145, 24, 0, 0], [-42, 42, 0, 0], [125, 30, 0, 0], [0, 0, -18, -12], [-84, -12, 0, 0], [0, 0, -67, 10], [-132, -36, 0, 0], [-170, -84, 0, 0], [0, 0, 108, 0], [50, -24, 0, 0], [0, 0, 14, -62], [0, 0, -108, 18], [-42, 90, 0, 0], [0, 0, 59, 44], [-126, -42, 0, 0], [109, 84, 0, 0], [72, -138, 0, 0], [0, 0, -36, -28], [-74, -36, 0, 0], [0, 0, -132, 0], [0, 0, 30, -48], [0, 0, -37, -12], [246, 6, 0, 0], [-282, 60, 0, 0], [0, 0, -96, -104], [0, 0, 20, -74], [0, 0, 6, 6], [-96, -132, 0, 0], [0, 0, 130, -64], [0, 0, 60, 8], [-205, 48, 0, 0], [96, 162, 0, 0], [0, 0, 19, -66], [0, 0, -20, -112], [0, 0, -19, -234], [0, 0, -58, 124], [0, 0, 155, 56], [60, 24, 0, 0], [-185, -54, 0, 0], [289, -108, 0, 0], [144, 126, 0, 0], [0, 0, 36, -116], [0, 0, 50, -218], [-156, 96, 0, 0], [0, 0, -211, -22], [-325, -252, 0, 0], [-209, -294, 0, 0], [0, 0, -104, -130], [606, -30, 0, 0], [0, 0, 109, -108], [420, -102, 0, 0], [0, 0, -229, 16], [0, 0, -58, -164], [-85, -84, 0, 0], [-54, 318, 0, 0], [0, 0, 229, 180], [0, 0, -60, 120], [144, -18, 0, 0], [-366, -246, 0, 0], [-227, 72, 0, 0], [0, 0, 40, 296], [233, 450, 0, 0], [0, 0, -78, 54], [0, 0, -188, -22], [7, -402, 0, 0], [84, 228, 0, 0], [-329, -30, 0, 0], [0, 0, -26, 8], [0, 0, 58, -370], [0, 0, 328, 80], [0, 0, 235, -94], [119, 72, 0, 0], [26, -132, 0, 0], [678, -144, 0, 0], [0, 0, -78, -258], [-378, -288, 0, 0], [125, -594, 0, 0], [0, 0, 299, -104], [0, 0, 252, 0], [0, 0, 442, -28], [-636, -168, 0, 0], [0, 0, 47, -48], [0, 0, -55, 26], [-562, -108, 0, 0], [-294, 660, 0, 0], [0, 0, 187, 62], [-598, -384, 0, 0], [0, -162, 0, 0], [0, 0, 451, 30], [0, 0, -630, 36], [-558, 504, 0, 0], [96, -486, 0, 0], [0, 0, 71, -16], [229, -132, 0, 0], [0, 0, 372, 60], [-444, -546, 0, 0], [0, 0, 19, -426], [348, 246, 0, 0], [730, -336, 0, 0], [0, 0, -38, 428], [0, 0, 67, 210], [0, 0, -359, 132], [269, -54, 0, 0], [114, 564, 0, 0], [-485, 66, 0, 0], [526, -480, 0, 0], [0, 0, -104, -22], [0, 0, -479, -188], [0, 0, 290, -368], [0, 0, -384, -364], [0, 0, 372, -18], [102, -90, 0, 0], [0, 0, 288, -420], [210, -66, 0, 0], [-502, 444, 0, 0], [798, 450, 0, 0], [0, 0, 431, 320], [0, 0, -64, -44], [0, 0, -265, -124], [0, 0, 728, -140], [0, 0, 72, 308], [-690, -240, 0, 0], [326, 228, 0, 0], [0, 0, -36, -174], [187, 954, 0, 0], [0, 0, 350, -212], [-113, 246, 0, 0], [480, -498, 0, 0], [-737, -138, 0, 0], [0, 0, -466, -38], [0, 0, -565, 304], [0, 0, -656, -70], [-162, -912, 0, 0], [-528, -438, 0, 0], [-343, 534, 0, 0], [0, 0, 328, -178], [570, 720, 0, 0], [0, 0, -716, -166], [-281, 246, 0, 0], [0, 0, 133, -504]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_3_c_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_882_3_c_b(:prec:=4)
    chi := MakeCharacter_882_c();
    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_882_3_c_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_882_3_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![144, 0, 72, 0, 1]>,<23,R![-504, -24, 1]>],Snew);
    return Vf;
end function;