
// Make newform 35.4.e.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_35_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_35_e_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_35_4_e_a();".
// 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_35_4_e_a();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_35_e();"
function MakeCharacter_35_e()
    N := 35;
    order := 3;
    char_gens := [22, 31];
    v := [3, 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_35_e_Hecke();"
function MakeCharacter_35_e_Hecke(Kf)
    N := 35;
    order := 3;
    char_gens := [22, 31];
    char_values := [[1, 0], [0, -1]];
    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, -3], [2, -2], [0, -5], [-7, -14], [45, -45], [59, 0], [54, -54], [0, 121], [0, -69], [-162, 0], [88, -88], [0, 259], [195, 0], [-286, 0], [0, -45], [-597, 597], [360, -360], [0, -392], [280, -280], [48, 0], [-668, 668], [0, -782], [768, 0], [0, 1194], [902, 0], [-684, 684], [0, 1516], [0, 732], [1600, -1600], [-1392, 0], [803, 0], [0, -2019], [-60, 60], [-1708, 0], [0, 1086], [2866, -2866], [229, -229], [0, 1228], [-1929, 0], [0, 699], [-3117, 3117], [-1798, 0], [0, 2388], [-272, 272], [-2109, 0], [-1424, 1424], [-3625, 0], [-4960, 0], [1500, -1500], [0, -6092], [0, -138], [-5502, 0], [-3551, 3551], [7065, 0], [0, 4080], [3288, -3288], [3264, -3264], [0, 2752], [4690, -4690], [7821, 0], [658, -658], [-5997, 0], [-6226, 0], [-4680, 4680], [0, -1028], [0, -8622], [0, 1999], [5114, 0], [-4320, 4320], [7922, 0], [-828, 828], [0, 1350], [-2801, 2801], [0, -6602], [-8305, 0], [0, -945], [-12036, 12036], [0, 2698], [0, -7053], [10870, -10870], [-9729, 0], [-12550, 0], [-2988, 2988], [16616, 0], [0, -7346], [0, -12], [9669, 0], [0, 9634], [-342, 0], [2411, 0], [0, 1206], [432, -432], [11896, -11896], [-12276, 0], [0, 10876], [12000, 0], [0, 11682], [-9609, 9609], [0, -21188], [0, -8072], [344, 0], [-18363, 18363], [6294, -6294], [0, -11733], [-1052, 1052], [13156, -13156], [-13368, 0], [0, -26664], [-7614, 7614], [6410, 0], [0, 21469], [-3737, 3737], [18078, 0], [-12287, 12287], [-9580, 0], [-10779, 10779], [8882, 0], [11019, -11019], [0, -22323], [-11856, 0], [33244, -33244], [-12322, 0], [0, 12597], [8340, -8340], [0, 20200], [474, 0], [0, 25126], [0, 7296], [-15421, 0], [0, 29167], [13381, -13381], [5487, 0], [0, -6638], [14846, 0], [0, 3651], [29855, 0], [6519, -6519], [-35114, 35114], [20910, 0], [-4431, 4431], [-9577, 0], [0, 10938], [-11540, 11540], [-18762, 0], [39610, -39610], [39162, 0], [-11527, 0], [41826, -41826], [0, -35192], [0, 9063], [0, -28439], [-9303, 0], [-14728, 0], [0, 17016], [24922, -24922], [30714, 0], [0, -17426], [0, -26649], [27686, 0], [17808, -17808], [0, 6906], [-20940, 0], [9176, 0], [0, -29763], [38490, -38490], [12609, -12609], [-19820, 19820], [-46034, 46034]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_35_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_35_4_e_a();".
// 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_35_4_e_a(:prec:=2)
    chi := MakeCharacter_35_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_35_4_e_a();".
// 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_35_4_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_35_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, 3, 1]>],Snew);
    return Vf;
end function;