
// Make newform 882.3.b.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_b_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_b_f();".
// 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_b_f();".
// 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 := [9, 0, 8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 5, 0, 1], [0, 22, 0, 2], [16, 0, 4, 0]];
    Rf_basisdens := [1, 3, 3, 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_b();"
function MakeCharacter_882_b()
    N := 882;
    order := 2;
    char_gens := [785, 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_882_b_Hecke();"
function MakeCharacter_882_b_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, 1, -1, 0], [0, 0, 0, 0], [0, -2, 2, 0], [8, 0, 0, 1], [0, -13, 1, 0], [-20, 0, 0, 0], [0, -2, 2, 0], [0, -19, -2, 0], [-4, 0, 0, -2], [38, 0, 0, 0], [0, -27, -3, 0], [20, 0, 0, 6], [0, -12, 0, 0], [0, -3, -12, 0], [0, -20, -4, 0], [-58, 0, 0, 4], [-48, 0, 0, -8], [0, -2, 2, 0], [24, 0, 0, 5], [76, 0, 0, -4], [0, -64, 4, 0], [0, -51, 9, 0], [72, 0, 0, 11], [0, 49, 5, 0], [-68, 0, 0, -6], [0, -90, 6, 0], [-72, 0, 0, 10], [0, -1, 4, 0], [172, 0, 0, -4], [0, -20, -16, 0], [0, 65, -2, 0], [-48, 0, 0, -16], [0, -51, -12, 0], [-72, 0, 0, -2], [58, 0, 0, 12], [-40, 0, 0, 12], [0, 64, -4, 0], [0, 25, 17, 0], [0, -126, -6, 0], [-136, 0, 0, -5], [0, 66, 18, 0], [134, 0, 0, 0], [0, 91, 8, 0], [88, 0, 0, -18], [-148, 0, 0, 6], [-32, 0, 0, 18], [0, 72, 0, 0], [216, 0, 0, 5], [0, -119, 26, 0], [0, 30, -18, 0], [-248, 0, 0, -9], [0, 252, 0, 0], [0, -69, -21, 0], [0, -90, -18, 0], [0, -119, 35, 0], [20, 0, 0, -2], [32, 0, 0, 0], [0, -65, -10, 0], [-228, 0, 0, 20], [0, -9, 51, 0], [108, 0, 0, 8], [0, -140, 44, 0], [126, 0, 0, -16], [0, 51, 24, 0], [44, 0, 0, 30], [-320, 0, 0, 2], [0, -114, -6, 0], [-138, 0, 0, -28], [0, -147, 3, 0], [0, 122, 46, 0], [-200, 0, 0, 30], [-318, 0, 0, -8], [420, 0, 0, -2], [0, 188, 64, 0], [0, 389, -2, 0], [-202, 0, 0, 28], [0, 105, -6, 0], [232, 0, 0, -21], [0, 240, 0, 0], [-176, 0, 0, -40], [0, -90, -78, 0], [130, 0, 0, -48], [128, 0, 0, -62], [0, 154, -34, 0], [0, -121, 76, 0], [48, 0, 0, 28], [0, -257, -1, 0], [36, 0, 0, 4], [0, -452, 32, 0], [0, -328, -32, 0], [-400, 0, 0, 30], [0, 162, -6, 0], [12, 0, 0, 70], [0, -24, -84, 0], [0, 687, -3, 0], [0, 267, 45, 0], [232, 0, 0, 0], [208, 0, 0, -4], [592, 0, 0, -28], [0, -139, 16, 0], [0, 380, 64, 0], [0, 129, 54, 0], [232, 0, 0, 68], [318, 0, 0, -16], [0, -40, 52, 0], [0, -21, 33, 0], [0, 90, 6, 0], [418, 0, 0, 0], [-352, 0, 0, -26], [-110, 0, 0, -16], [0, -137, -22, 0], [264, 0, 0, 68], [-172, 0, 0, -12], [0, 337, -22, 0], [604, 0, 0, 0], [0, 0, 24, 0], [0, -309, -6, 0], [0, -758, 26, 0], [-198, 0, 0, 68], [-406, 0, 0, -24], [0, 417, 57, 0], [0, 198, -6, 0], [-584, 0, 0, 60], [0, 101, -38, 0], [8, 0, 0, -6], [0, -396, -60, 0], [852, 0, 0, -46], [152, 0, 0, 19], [880, 0, 0, -52], [0, -270, 66, 0], [88, 0, 0, -22], [184, 0, 0, -10], [0, 269, 145, 0], [-306, 0, 0, -40], [0, 79, -73, 0], [608, 0, 0, -56], [0, -87, -123, 0], [0, -313, -80, 0], [48, 0, 0, -52], [0, 355, -64, 0], [-516, 0, 0, -8], [0, 894, 6, 0], [696, 0, 0, -7], [0, 884, -8, 0], [-518, 0, 0, 36], [0, -99, 81, 0], [484, 0, 0, -48], [0, 74, -14, 0], [66, 0, 0, 88], [0, -21, 129, 0], [-724, 0, 0, -78], [0, -696, 12, 0], [-284, 0, 0, 14], [0, 402, 90, 0], [-240, 0, 0, -14], [0, 243, -129, 0], [-1262, 0, 0, 0], [0, 703, -43, 0], [0, -238, -110, 0], [0, 73, 8, 0], [1268, 0, 0, -36], [0, -684, -60, 0], [0, -81, -30, 0], [0, 36, -72, 0], [-112, 0, 0, -78], [-842, 0, 0, 36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_3_b_f();".
// 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_b_f(:prec:=4)
    chi := MakeCharacter_882_b();
    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_b_f();".
// 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_b_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![2916, 0, 116, 0, 1]>,<11,R![46656, 0, 464, 0, 1]>,<13,R![-48, -16, 1]>],Snew);
    return Vf;
end function;