
// Make newform 252.4.k.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_252_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_252_k_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_252_4_k_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_252_4_k_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 := [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_252_k();"
function MakeCharacter_252_k()
    N := 252;
    order := 3;
    char_gens := [127, 29, 73];
    v := [3, 3, 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_252_k_Hecke();"
function MakeCharacter_252_k_Hecke(Kf)
    N := 252;
    order := 3;
    char_gens := [127, 29, 73];
    char_values := [[1, 0], [1, 0], [-1, 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, 0], [0, 0], [0, 0], [-1, 19], [0, 0], [-19, 0], [0, 0], [-107, 107], [0, 0], [0, 0], [0, 289], [-323, 323], [0, 0], [71, 0], [0, 0], [0, 0], [0, 0], [-182, 182], [0, 127], [0, 0], [0, 271], [1387, -1387], [0, 0], [0, 0], [-1330, 0], [0, 0], [1801, -1801], [0, 0], [0, -2213], [0, 0], [2267, 0], [0, 0], [0, 0], [467, 0], [0, 0], [0, -1748], [0, 3850], [3400, -3400], [0, 0], [0, 0], [0, 0], [-4699, 0], [0, 0], [0, 3961], [0, 0], [0, 5236], [6032, 0], [-3220, 0], [0, 0], [-2357, 2357], [0, 0], [0, 0], [0, 7378], [0, 0], [0, 0], [0, 0], [0, 0], [-812, 812], [0, -9197], [0, 0], [0, 9469], [0, 0], [-3943, 0], [0, 0], [9109, -9109], [0, 0], [10891, -10891], [12293, 0], [0, 0], [-11914, 0], [0, 0], [0, 0], [0, -9413], [12601, -12601], [-6103, 0], [0, 0], [0, 0], [14257, -14257], [0, 0], [0, 16543], [0, 0], [-6679, 0], [0, 0], [16739, 0], [-14924, 14924], [0, 0], [0, 0], [-6497, 6497], [0, 0], [11969, 0], [0, 0], [0, 0], [0, 6103], [0, 0], [-21743, 21743], [0, 0], [0, 0], [0, 0], [-23921, 23921], [-1889, 1889], [1640, 0], [0, 0], [0, 0], [0, 0], [0, 23941], [0, 9613], [0, 0], [0, 0], [0, 0], [17351, 0], [-6137, 6137], [0, -17390], [0, 0], [0, -26693], [1892, 0], [0, 0], [19259, 0], [0, 0], [0, 0], [0, 0], [0, -33731], [9899, 0], [0, 0], [0, 0], [20179, -20179], [0, 0], [-36146, 36146], [0, 0], [38033, 0], [39331, -39331], [0, -6047], [0, 0], [17569, -17569], [-41470, 0], [0, 0], [-34417, 0], [0, 0], [0, -43400], [0, 0], [0, 0], [39368, 0], [0, 0], [0, 12220], [0, 0], [0, -30077], [0, 0], [8279, 0], [0, 0], [-31304, 31304], [0, 0], [-50150, 50150], [0, 0], [52109, 0], [0, 0], [0, 49447], [0, 0], [-46817, 46817], [0, 0], [40283, 0], [0, 0], [0, 0], [0, 0], [-3907, 0], [0, 0], [0, 0], [0, 0], [0, 14041], [0, 62893]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_252_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_252_4_k_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_252_4_k_b(:prec:=2)
    chi := MakeCharacter_252_k();
    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_252_4_k_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_252_4_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_252_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![19, 1]>],Snew);
    return Vf;
end function;