
// Make newform 240.3.c.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_240_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_240_c_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_c_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_240_3_c_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, 0, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 4, 0, 0], [3, 4, 2, 2], [3, -4, 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_c();"
function MakeCharacter_240_c()
    N := 240;
    order := 2;
    char_gens := [31, 181, 161, 97];
    v := [2, 2, 1, 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_240_c_Hecke();"
function MakeCharacter_240_c_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, 0, 1], [0, 1, 0, -1], [0, 0, -2, 2], [0, -2, -1, 1], [0, 0, -3, 3], [0, 0, 7, 7], [6, 0, 0, 0], [0, 0, 1, 1], [0, 6, 3, -3], [-34, 0, 0, 0], [0, 0, 11, -11], [0, -4, -2, 2], [0, 0, -7, 7], [0, 0, 1, 1], [0, 0, -9, -9], [0, 22, 11, -11], [74, 0, 0, 0], [0, 0, 23, -23], [0, 12, 6, -6], [0, 0, -14, 14], [78, 0, 0, 0], [0, 0, -23, -23], [0, 4, 2, -2], [0, 0, -8, 8], [0, 34, 17, -17], [0, 0, -26, 26], [0, 0, 33, 33], [74, 0, 0, 0], [0, 0, -9, -9], [0, 0, 4, -4], [0, 18, 9, -9], [0, 0, 39, 39], [118, 0, 0, 0], [0, -22, -11, 11], [-34, 0, 0, 0], [0, 0, -23, 23], [0, 0, -17, 17], [0, 0, -15, -15], [0, 0, -17, -17], [0, 58, 29, -29], [-166, 0, 0, 0], [0, -48, -24, 24], [0, 0, -8, 8], [0, 0, -1, -1], [-114, 0, 0, 0], [6, 0, 0, 0], [0, 0, 68, -68], [0, 0, -55, -55], [154, 0, 0, 0], [0, 0, -41, -41], [0, -40, -20, 20], [-206, 0, 0, 0], [0, 6, 3, -3], [0, 0, -9, -9], [0, 0, -47, -47], [0, 30, 15, -15], [398, 0, 0, 0], [0, 0, -73, 73], [0, -12, -6, 6], [0, 0, 13, -13], [0, 0, 87, 87], [0, 0, 123, -123], [0, -108, -54, 54], [0, 0, 142, -142], [0, 0, -137, -137], [-202, 0, 0, 0], [0, 0, 92, -92], [0, 0, 57, 57], [-118, 0, 0, 0], [0, 0, 7, 7], [0, 12, 6, -6], [0, 0, 88, -88], [0, 0, -33, 33], [-394, 0, 0, 0], [0, 0, 17, 17], [0, -62, -31, 31], [0, 0, -163, 163], [0, 40, 20, -20], [-206, 0, 0, 0], [0, -46, -23, 23], [-38, 0, 0, 0], [0, 136, 68, -68], [0, 0, -68, 68], [366, 0, 0, 0], [0, 0, 129, 129], [0, 96, 48, -48], [0, 0, 26, -26], [0, -114, -57, 57], [0, 0, 24, -24], [0, 0, 33, 33], [0, 128, 64, -64], [0, 0, -162, 162], [0, 30, 15, -15], [486, 0, 0, 0], [0, 0, 177, 177], [0, 6, 3, -3], [0, 220, 110, -110], [0, 0, 73, -73], [-86, 0, 0, 0], [0, 0, -41, 41], [0, 0, -81, -81], [0, 0, -223, -223], [0, -140, -70, 70], [-394, 0, 0, 0], [0, 0, 152, -152], [0, 0, -159, -159], [0, 0, 135, 135], [0, -12, -6, 6], [434, 0, 0, 0], [0, 0, 164, -164], [0, 0, 211, -211], [0, 0, 103, 103], [1046, 0, 0, 0], [46, 0, 0, 0], [0, -256, -128, 128], [0, 0, -201, 201], [0, 0, 129, 129], [0, 0, -241, -241], [0, -182, -91, 91], [1082, 0, 0, 0], [0, 0, -264, 264], [0, 0, -137, -137], [0, 0, -71, -71], [-922, 0, 0, 0], [0, -106, -53, 53], [-966, 0, 0, 0], [0, 248, 124, -124], [0, 0, -102, 102], [0, 0, 41, -41], [-1082, 0, 0, 0], [0, 0, -255, -255], [958, 0, 0, 0], [0, 0, -193, 193], [0, -252, -126, 126], [-1326, 0, 0, 0], [0, 0, -33, -33], [0, 0, 283, -283], [0, 0, -137, -137], [0, -276, -138, 138], [-874, 0, 0, 0], [0, 58, 29, -29], [0, 0, -342, 342], [0, 0, -103, -103], [1002, 0, 0, 0], [0, -68, -34, 34], [0, 0, -193, 193], [0, 0, -201, -201], [278, 0, 0, 0], [0, 0, 129, 129], [0, 0, 157, -157], [0, 120, 60, -60], [0, 0, -237, 237], [0, 0, -63, -63], [0, 0, 129, -129], [0, -72, -36, 36], [-1314, 0, 0, 0], [0, 336, 168, -168], [0, 0, 322, -322], [0, -82, -41, 41], [0, 0, -375, -375], [0, 0, -185, -185], [0, 0, 134, -134], [0, 30, 15, -15], [0, 0, 39, 39], [0, 0, -415, -415], [1646, 0, 0, 0], [0, 0, 291, -291]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_240_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_240_3_c_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_240_3_c_d(:prec:=4)
    chi := MakeCharacter_240_c();
    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_c_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_240_3_c_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_240_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![64, 0, 1]>,<17,R![-980, 0, 1]>],Snew);
    return Vf;
end function;