
// Make newform 882.3.c.e in Magma, downloaded from the LMFDB on 25 April 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_e();".
// 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_e();".
// 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 := [2, 0, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [2, 0, 1, 0], [0, 3, 0, 1]];
    Rf_basisdens := [1, 1, 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_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, 0, -1, 0], [0, 0, 0, 0], [0, 1, 0, 2], [0, 0, 0, 0], [0, 0, -2, 0], [0, -5, 0, -8], [0, -6, 0, 9], [0, -6, 0, 10], [28, 0, 6, 0], [28, 0, -11, 0], [0, 14, 0, -14], [16, 0, 21, 0], [0, 35, 0, 14], [-32, 0, -28, 0], [0, -10, 0, 50], [-42, 0, -44, 0], [0, -46, 0, 6], [0, -37, 0, 8], [4, 0, 0, 0], [112, 0, -14, 0], [0, 32, 0, 33], [-36, 0, 28, 0], [0, 76, 0, 12], [0, -30, 0, 73], [0, 64, 0, -39], [0, -62, 0, -19], [0, 34, 0, 74], [28, 0, -10, 0], [8, 0, 91, 0], [-14, 0, -84, 0], [-8, 0, 112, 0], [0, 72, 0, 88], [-28, 0, -17, 0], [0, 72, 0, 48], [70, 0, -36, 0], [-4, 0, 112, 0], [0, -8, 0, 67], [-12, 0, 140, 0], [0, -110, 0, -66], [0, -23, 0, -102], [112, 0, 34, 0], [0, 104, 0, 11], [-112, 0, 30, 0], [56, 0, 56, 0], [-112, 0, 64, 0], [0, 156, 0, 76], [-212, 0, -28, 0], [0, 20, 0, -24], [0, 118, 0, -198], [0, -144, 0, 37], [-196, 0, -15, 0], [-196, 0, 90, 0], [0, -73, 0, 40], [0, -78, 0, -58], [0, 117, 0, -214], [-112, 0, 86, 0], [0, 234, 0, -43], [0, 162, 0, 206], [314, 0, 0, 0], [-84, 0, 169, 0], [0, 194, 0, -230], [0, -274, 0, -149], [0, -226, 0, 162], [0, 54, 0, 206], [0, -296, 0, -111], [-336, 0, 56, 0], [116, 0, 28, 0], [208, 0, -119, 0], [364, 0, 202, 0], [0, 56, 0, -357], [0, -3, 0, -286], [-252, 0, -198, 0], [0, -144, 0, -152], [152, 0, -56, 0], [-308, 0, 196, 0], [0, 332, 0, 104], [140, 0, 85, 0], [0, 53, 0, -352], [420, 0, -193, 0], [0, -104, 0, 129], [0, -178, 0, 386], [358, 0, -280, 0], [-392, 0, -278, 0], [0, 201, 0, 232], [0, 204, 0, -88], [-56, 0, 246, 0], [350, 0, 76, 0], [32, 0, -504, 0], [0, 383, 0, -74], [-144, 0, 364, 0], [0, -38, 0, -6], [0, 74, 0, -146], [-804, 0, -28, 0], [84, 0, -494, 0], [264, 0, 280, 0], [0, 292, 0, -284], [0, 58, 0, 445], [0, -395, 0, -174], [0, 524, 0, 200], [-400, 0, -448, 0], [448, 0, 336, 0], [-336, 0, -32, 0], [0, -410, 0, 286], [-252, 0, 271, 0], [176, 0, -196, 0], [0, -681, 0, -216], [0, 522, 0, 106], [0, 38, 0, 223], [392, 0, -454, 0], [0, 72, 0, -351], [0, 340, 0, -324], [360, 0, 315, 0], [252, 0, -191, 0], [0, 216, 0, -332], [-16, 0, 504, 0], [-588, 0, -281, 0], [0, -550, 0, -138], [0, -142, 0, 122], [1036, 0, -67, 0], [840, 0, 226, 0], [0, -304, 0, -275], [-88, 0, -343, 0], [0, -193, 0, 6], [-336, 0, 426, 0], [0, -152, 0, 132], [84, 0, -445, 0], [-280, 0, 77, 0], [0, -168, 0, 196], [0, -290, 0, 614], [0, -424, 0, -5], [-284, 0, -252, 0], [-448, 0, -214, 0], [400, 0, 280, 0], [280, 0, -301, 0], [0, 574, 0, 231], [0, -177, 0, -496], [0, -447, 0, -278], [0, 220, 0, -852], [0, -570, 0, 99], [672, 0, 48, 0], [0, -236, 0, -316], [-448, 0, 232, 0], [-228, 0, -56, 0], [-252, 0, 542, 0], [0, 11, 0, 432], [0, -134, 0, 110], [0, 192, 0, -5], [0, 278, 0, -417], [0, -414, 0, 1026], [-196, 0, 510, 0], [728, 0, -539, 0], [0, 2, 0, -199], [216, 0, 196, 0], [0, 446, 0, -74], [576, 0, -504, 0], [-56, 0, 518, 0], [-1260, 0, 168, 0], [0, -806, 0, -639], [0, 264, 0, -377], [0, 6, 0, 19], [-1204, 0, -222, 0], [-1456, 0, -40, 0], [-48, 0, 252, 0], [0, 160, 0, -436], [252, 0, 769, 0], [0, 88, 0, -552], [-276, 0, 1120, 0], [0, -16, 0, 141]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![2, 0, 20, 0, 1]>,<23,R![712, -56, 1]>],Snew);
    return Vf;
end function;