
// Make newform 240.3.e.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_240_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_240_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_240_3_e_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_240_3_e_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, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [2, 0, 0, 1], [0, -2, 2, -1], [1, 6, -2, 2]];
    Rf_basisdens := [1, 1, 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_240_e();"
function MakeCharacter_240_e()
    N := 240;
    order := 2;
    char_gens := [31, 181, 161, 97];
    v := [1, 2, 2, 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_240_e_Hecke();"
function MakeCharacter_240_e_Hecke(Kf)
    N := 240;
    order := 2;
    char_gens := [31, 181, 161, 97];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [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, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [0, 0, -2, -2], [0, 0, -2, -2], [4, -6, 0, 0], [-12, -6, 0, 0], [0, 0, 8, -4], [0, 0, 4, 4], [12, -18, 0, 0], [0, 0, -8, -4], [-40, 6, 0, 0], [42, 0, 0, 0], [0, 0, 4, -16], [0, 0, 8, 8], [-48, 18, 0, 0], [0, 0, 22, -2], [86, 12, 0, 0], [0, 0, 16, -12], [0, 0, 4, 28], [22, 36, 0, 0], [0, 0, 0, 4], [0, 0, -48, 12], [-66, 12, 0, 0], [-86, 24, 0, 0], [180, 6, 0, 0], [0, 0, 54, 14], [0, 0, 80, -4], [-74, -12, 0, 0], [-24, -42, 0, 0], [0, 0, 54, 6], [0, 0, -14, -38], [-48, -42, 0, 0], [0, 0, 12, -40], [-48, 42, 0, 0], [0, 0, -76, 8], [-28, -78, 0, 0], [0, 0, -16, 12], [0, 0, 92, -28], [-120, -30, 0, 0], [0, 0, 118, 22], [46, -132, 0, 0], [0, 0, 156, -12], [134, -12, 0, 0], [96, -30, 0, 0], [0, 0, -76, 48], [0, 0, -4, 96], [0, 0, 166, -10], [0, 0, -20, 64], [10, -72, 0, 0], [-156, 18, 0, 0], [0, 0, -52, -76], [-182, 120, 0, 0], [0, 0, 58, -14], [192, 126, 0, 0], [0, 0, -68, 52], [228, 102, 0, 0], [0, 0, 12, -32], [184, -66, 0, 0], [-114, 12, 0, 0], [0, 0, 32, 76], [-144, 162, 0, 0], [0, 0, -124, 64], [0, 0, 340, 4], [182, 36, 0, 0], [-72, 42, 0, 0], [0, 0, -96, -28], [238, -36, 0, 0], [0, 0, 36, -48], [134, -36, 0, 0], [-276, -6, 0, 0], [0, 0, -256, -16], [0, 0, -178, -42], [44, -102, 0, 0], [0, 0, 172, 0], [0, 0, -64, -136], [240, -150, 0, 0], [272, 126, 0, 0], [-138, 12, 0, 0], [130, 48, 0, 0], [0, 0, 18, -126], [-422, -168, 0, 0], [0, 0, 272, -40], [-178, 108, 0, 0], [0, 0, -164, -88], [0, 0, -428, 16], [-354, -156, 0, 0], [-14, -96, 0, 0], [-192, -246, 0, 0], [0, 0, 114, -118], [0, 0, -8, 100], [0, 0, -108, 180], [0, 0, 338, -6], [0, 0, -6, -150], [0, 0, -64, -20], [0, 0, 24, 96], [180, 102, 0, 0], [102, 60, 0, 0], [0, 0, 32, -76], [142, 204, 0, 0], [0, 0, -300, -40], [-648, 66, 0, 0], [0, 0, -172, 8], [-258, 276, 0, 0], [0, 0, -208, -28], [74, 408, 0, 0], [0, 0, 96, -84], [-288, -18, 0, 0], [0, 0, -24, -24], [-22, 72, 0, 0], [0, 0, -150, 2], [-184, 6, 0, 0], [-276, 234, 0, 0], [0, 0, 404, -112], [0, 0, -320, 12], [546, -216, 0, 0], [0, 0, -144, -36], [0, 0, 128, 152], [768, 186, 0, 0], [0, 0, -30, -6], [182, 204, 0, 0], [-10, -12, 0, 0], [-480, -150, 0, 0], [0, 0, -16, -268], [0, 0, 144, -12], [468, -186, 0, 0], [-302, 96, 0, 0], [0, 0, 32, 104], [0, 0, -182, 50], [-656, 246, 0, 0], [0, 0, 96, 12], [0, 0, 232, -152], [0, 0, 72, 172], [232, -354, 0, 0], [270, 180, 0, 0], [194, -432, 0, 0], [-456, 186, 0, 0], [0, 0, 296, 116], [-744, -222, 0, 0], [-654, -360, 0, 0], [0, 0, -316, -8], [228, 6, 0, 0], [0, 0, -110, -150], [0, 0, -44, 40], [-554, 372, 0, 0], [0, 0, 152, -184], [344, -282, 0, 0], [-180, 450, 0, 0], [0, 0, -228, 80], [0, 0, 84, -12], [620, 138, 0, 0], [-210, 12, 0, 0], [0, 0, -200, 52], [0, 0, 340, -164], [0, 0, 108, 40], [0, 0, -112, -160], [0, 0, -328, 148], [-258, -108, 0, 0], [-934, -168, 0, 0], [96, 234, 0, 0], [0, 0, -84, 120], [1416, 126, 0, 0], [0, 0, 582, -146], [0, 0, -902, -14], [1356, -222, 0, 0], [0, 0, 416, 176], [0, 0, 652, 136], [500, 546, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_240_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_240_3_e_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_240_3_e_b(:prec:=4)
    chi := MakeCharacter_240_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_240_3_e_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_240_3_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_240_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<13,R![-164, -8, 1]>],Snew);
    return Vf;
end function;