
// Make newform 1008.4.b.d in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_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_1008_4_b_d();".
// 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_1008_4_b_d();".
// 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], [-9, 18]];
    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_1008_b();"
function MakeCharacter_1008_b()
    N := 1008;
    order := 2;
    char_gens := [127, 757, 785, 577];
    v := [1, 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_1008_b_Hecke();"
function MakeCharacter_1008_b_Hecke(Kf)
    N := 1008;
    order := 2;
    char_gens := [127, 757, 785, 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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, 0], [10, -1], [0, 0], [0, -4], [0, 0], [56, 0], [0, 0], [0, 0], [-308, 0], [-110, 0], [0, 0], [0, -14], [0, 0], [0, 0], [0, 0], [0, 60], [0, -42], [0, 0], [0, 24], [0, -70], [0, 0], [0, 0], [0, -88], [0, 0], [-1820, 0], [0, 0], [-646, 0], [0, 0], [0, -182], [0, 0], [0, 0], [-2576, 0], [0, 0], [0, -210], [0, -52], [0, -154], [0, 0], [0, 0], [0, 0], [0, 220], [0, 0], [1150, 0], [0, 0], [-5236, 0], [0, -70], [-3220, 0], [0, 0], [0, -340], [0, 0], [0, 0], [0, 80], [0, 0], [0, 0], [0, 0], [0, 0], [-812, 0], [4030, 0], [0, 0], [5600, 0], [0, 0], [10640, 0], [0, 0], [0, -304], [0, 0], [0, 770], [-4930, 0], [0, 0], [0, 340], [0, 0], [0, 0], [4340, 0], [12350, 0], [0, 770], [0, 0], [0, 0], [0, 1012], [0, 0], [0, 920], [0, 0], [-17138, 0], [0, 0], [0, -1144], [14924, 0], [0, 0], [0, 0], [12710, 0], [0, 0], [0, 154], [0, 0], [0, 0], [0, -322], [0, 0], [0, 1050], [0, 0], [0, 0], [0, 0], [12040, 0], [22678, 0], [0, 1638], [0, 0], [0, 0], [0, 0], [0, 910], [0, -1368], [0, 0], [0, 0], [0, 0], [0, 200], [-28420, 0], [17390, 0], [0, 0], [26656, 0], [0, -2030], [0, 0], [13160, 0], [0, 0], [0, 0], [0, 0], [0, 1740], [-24050, 0], [0, 0], [0, 0], [16072, 0], [0, 0], [36146, 0], [0, 0], [10780, 0], [0, -2356], [0, -1610], [0, 0], [0, -2170], [-41470, 0], [0, 0], [0, 2720], [0, 0], [-43400, 0], [0, 0], [0, 0], [-39368, 0], [0, 0], [0, -2926], [0, 0], [0, -2860], [0, 0], [0, -1116], [0, 0], [31304, 0], [0, 0], [50150, 0], [0, 0], [0, -3094], [0, 0], [0, 2002], [0, 0], [0, -3570], [0, 0], [0, 928], [0, 0], [0, 0], [0, 0], [0, -2142], [0, 0], [0, 0], [0, 0], [0, -2730], [0, 3588], [-50654, 0], [0, 0], [0, 0], [0, -3300], [0, 0], [15370, 0], [21476, 0], [0, 0], [0, 2030], [0, 0], [-67340, 0], [0, 3700], [0, 4522], [0, 0], [-61490, 0], [0, 0], [0, 0], [0, 0], [56810, 0], [75040, 0], [73834, 0], [0, 0], [0, 4784], [0, 0], [0, -4550], [0, 0], [0, 0], [0, 0], [7198, 0], [56950, 0], [0, 0], [0, 0], [0, 0], [63308, 0], [0, -1628], [0, -4680], [0, 0], [0, 0], [85484, 0], [0, 0], [0, 0], [-41272, 0], [81650, 0], [0, 0], [0, 5166], [0, 0], [0, 0], [0, -3720], [0, 6118], [0, 0], [0, 0], [0, 0], [98642, 0], [-29036, 0], [0, 0], [0, 4774], [0, 0], [-53534, 0], [0, 0], [0, 0], [109060, 0], [0, 0], [-8710, 0], [-69496, 0], [0, 2870], [0, 0], [113960, 0], [0, 0], [0, 1480], [0, 0], [0, 0], [0, 0], [0, 0], [-31808, 0], [100100, 0], [49786, 0], [0, 0], [0, 0], [121660, 0], [0, 0], [0, 2590], [0, 0], [-114550, 0], [0, 0], [0, 0], [0, 6240], [0, 0], [0, 0], [108862, 0], [121520, 0], [0, 0], [0, -8648], [0, 8554], [0, 0], [0, -8460], [0, -2236], [0, 0], [-64064, 0], [0, 0], [0, 0], [0, 574], [0, 0], [0, 2860], [0, -7238], [0, 8816], [0, 3990], [0, -8432], [-127820, 0], [0, 0], [111274, 0], [-5402, 0], [0, 0], [0, 0], [0, -9310], [0, 0], [0, -2460], [138040, 0], [0, 0], [-90610, 0], [0, 0], [-22484, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-121210, 0], [0, 0], [5852, 0], [0, 0], [0, 0], [-174440, 0], [0, 10336], [0, 0], [0, 3290]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_4_b_d();".
// 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_1008_4_b_d(:prec:=2)
    chi := MakeCharacter_1008_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1008_4_b_d();".
// 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_1008_4_b_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<19,R![-56, 1]>],Snew);
    return Vf;
end function;