
// Make newform 152.3.e.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_152_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_152_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_152_3_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_152_3_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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 4]];
    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_152_e();"
function MakeCharacter_152_e()
    N := 152;
    order := 2;
    char_gens := [39, 77, 97];
    v := [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_152_e_Hecke();"
function MakeCharacter_152_e_Hecke(Kf)
    N := 152;
    order := 2;
    char_gens := [39, 77, 97];
    char_values := [[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 := 3;
    raw_aps := [[0, 0], [0, 1], [7, 0], [11, 0], [3, 0], [0, -2], [-17, 0], [-19, 0], [2, 0], [0, -7], [0, 1], [0, -7], [0, 7], [-21, 0], [-5, 0], [0, 1], [0, 6], [23, 0], [0, 7], [0, 16], [39, 0], [0, -17], [-6, 0], [0, 21], [0, -30], [122, 0], [0, 18], [0, 28], [0, -21], [0, 9], [0, -7], [-149, 0], [95, 0], [155, 0], [63, 0], [0, -22], [-150, 0], [-166, 0], [0, 37], [0, 10], [0, -13], [0, -14], [-301, 0], [0, 27], [90, 0], [147, 0], [0, -58], [0, 63], [0, -56], [-257, 0], [-177, 0], [363, 0], [0, 63], [-133, 0], [0, -24], [-101, 0], [0, 63], [-142, 0], [199, 0], [0, 82], [427, 0], [0, -42], [0, -93], [235, 0], [530, 0], [0, -38], [0, 17], [0, -30], [-253, 0], [351, 0], [98, 0], [-61, 0], [-686, 0], [0, 24], [0, 14], [0, -3], [599, 0], [-569, 0], [0, -3], [0, 105], [-230, 0], [0, -105], [0, -84], [0, 105], [0, 21], [-653, 0], [0, -26], [-817, 0], [463, 0], [-29, 0], [19, 0], [-206, 0], [0, 8], [-246, 0], [-21, 0], [386, 0], [0, -84], [0, -139], [0, 70], [7, 0], [0, 126], [-1001, 0], [0, -57], [0, -126], [746, 0], [-25, 0], [-869, 0], [898, 0], [0, -49], [0, -84], [0, -80], [-585, 0], [-873, 0], [970, 0], [259, 0], [0, 188], [-645, 0], [-29, 0], [135, 0], [0, 138], [0, -176], [0, -70], [0, 63], [0, 92], [835, 0], [1002, 0], [250, 0], [1171, 0], [91, 0], [682, 0], [-221, 0], [0, -51], [0, 164], [895, 0], [95, 0], [679, 0], [0, 0], [0, -246], [0, -63], [455, 0], [0, 84], [831, 0], [-109, 0], [0, 219], [0, -118], [0, 56], [-1126, 0], [0, -36], [651, 0], [0, 7], [0, 182], [391, 0], [995, 0], [0, -84], [0, -111], [0, -142], [1090, 0], [226, 0], [623, 0], [0, -274], [602, 0], [0, 49], [770, 0], [0, -126], [0, 237], [0, 98], [0, -113], [-473, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_152_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_152_3_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_152_3_e_a(:prec:=2)
    chi := MakeCharacter_152_e();
    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_152_3_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_152_3_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_152_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![32, 0, 1]>],Snew);
    return Vf;
end function;