
// Make newform 1560.2.w.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1560_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1560_w_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_1560_2_w_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_1560_2_w_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, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 0, 1], [0, 1, -1, 0]];
    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_1560_w();"
function MakeCharacter_1560_w()
    N := 1560;
    order := 2;
    char_gens := [391, 781, 521, 937, 1081];
    v := [2, 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_1560_w_Hecke();"
function MakeCharacter_1560_w_Hecke(Kf)
    N := 1560;
    order := 2;
    char_gens := [391, 781, 521, 937, 1081];
    char_values := [[1, 0, 0, 0], [-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 := 2;
    raw_aps := [[1, -1, 0, 0], [0, 0, -1, 0], [0, 0, 1, 0], [3, -1, 1, -1], [1, -1, 3, 1], [0, 0, 1, 0], [-4, 0, 0, 0], [1, -1, -1, 1], [-2, 0, 0, 0], [-4, 4, 2, -4], [-5, -3, 3, -3], [-2, 2, 4, -2], [2, 4, -4, 4], [-2, 2, 2, -2], [-7, 1, -1, 1], [-6, 6, -2, -6], [-5, 5, 1, -5], [4, -4, -4, 4], [-3, 3, -7, -3], [15, 1, -1, 1], [2, -8, 8, -8], [14, -2, 2, -2], [1, -1, -7, 1], [-6, -4, 4, -4], [-4, -2, 2, -2], [0, 0, 12, 0], [-6, 4, -4, 4], [0, 0, 4, 0], [-8, 8, -2, -8], [0, 0, 0, 0], [-10, 4, -4, 4], [-8, 8, 6, -8], [0, -6, 6, -6], [0, 0, 2, 0], [2, -2, 0, 2], [17, -1, 1, -1], [12, -12, 2, 12], [-3, 3, -15, -3], [15, 3, -3, 3], [-2, 2, -6, -2], [-6, 6, 0, -6], [4, -4, -2, 4], [-6, 6, -6, 6], [0, 2, -2, 2], [-6, 6, 8, -6], [-16, -4, 4, -4], [4, -4, -6, 4], [-1, 15, -15, 15], [-5, 5, 3, -5], [0, 0, 6, 0], [-2, -8, 8, -8], [-25, -3, 3, -3], [-14, 4, -4, 4], [6, -6, -4, 6], [-10, -4, 4, -4], [20, 2, -2, 2], [-8, 8, 18, -8], [-3, -5, 5, -5], [-2, 2, -18, -2], [-8, 6, -6, 6], [-2, 2, -2, -2], [12, -12, 2, 12], [-7, 7, -7, -7], [-4, 16, -16, 16], [8, 0, 0, 0], [-2, 2, 24, -2], [-11, 11, 7, -11], [-20, 0, 0, 0], [16, -16, 4, 16], [12, -12, 14, 12], [-6, -16, 16, -16], [15, 1, -1, 1], [-4, -2, 2, -2], [4, -4, -6, 4], [1, -1, 23, 1], [-19, -7, 7, -7], [4, -4, 2, 4], [-8, 8, -2, -8], [-10, 0, 0, 0], [0, 14, -14, 14], [20, -20, 2, 20], [10, -10, -12, 10], [-27, 7, -7, 7], [-10, -16, 16, -16], [-4, 8, -8, 8], [4, -4, 0, 4], [16, -6, 6, -6], [22, 4, -4, 4], [4, -4, -10, 4], [11, -5, 5, -5], [6, -6, -30, 6], [-7, 19, -19, 19], [-7, 1, -1, 1], [10, -10, 24, 10], [-3, 3, 3, -3], [4, -18, 18, -18], [-6, 6, 16, -6], [6, -2, 2, -2], [0, 0, -16, 0], [14, -14, 8, 14], [12, -12, -4, 12], [12, -12, 14, 12], [10, -10, -22, 10], [22, -8, 8, -8], [-6, 6, -16, -6], [-32, 6, -6, 6], [-5, 5, 11, -5], [12, 10, -10, 10], [-14, 14, -14, 14], [30, -6, 6, -6], [20, 2, -2, 2], [0, 0, 26, 0], [6, 4, -4, 4], [11, -11, 9, 11], [13, 3, -3, 3], [26, -2, 2, -2], [-17, 17, -5, -17], [-18, 12, -12, 12], [-4, 4, -10, -4], [0, 0, -34, 0], [-6, 6, 0, -6], [-16, -4, 4, -4], [-2, 2, 10, -2], [1, -1, -35, 1], [9, -9, 19, 9], [0, 0, 18, 0], [4, -4, -42, 4], [12, 8, -8, 8], [2, 4, -4, 4], [6, -6, -4, 6], [-9, 9, 29, -9], [-3, 17, -17, 17], [-16, 0, 0, 0], [6, -6, -22, 6], [-8, 6, -6, 6], [16, 10, -10, 10], [-18, 18, 0, -18], [-15, 15, -23, -15], [-16, 16, -6, -16], [10, -6, 6, -6], [-1, 1, 25, -1], [4, -4, 18, 4], [14, 0, 0, 0], [-5, 5, 31, -5], [0, 0, 28, 0], [11, -11, 11, -11], [12, -12, 6, 12], [18, -16, 16, -16], [4, -4, -6, 4], [-33, 7, -7, 7], [-6, 6, 28, -6], [6, 24, -24, 24], [10, -10, -14, 10], [-12, 18, -18, 18], [-10, 10, -22, -10], [10, 22, -22, 22], [-30, 2, -2, 2], [0, -6, 6, -6], [14, 20, -20, 20], [-2, 2, 8, -2], [-3, 3, -19, -3], [6, -4, 4, -4], [11, -1, 1, -1], [0, 0, 30, 0], [0, -22, 22, -22], [35, -13, 13, -13], [-2, 18, -18, 18], [22, -22, 22, 22], [48, -2, 2, -2], [-6, 6, -14, -6], [-17, 17, -11, -17], [-6, 6, 0, -6], [12, -16, 16, -16], [-10, -12, 12, -12], [8, 8, -8, 8], [38, -8, 8, -8], [-3, 3, 47, -3], [-18, 18, -16, -18], [-18, -12, 12, -12], [-24, 24, 10, -24], [-23, 1, -1, 1], [2, -2, 16, 2], [26, -26, -14, 26], [38, 12, -12, 12], [-17, -17, 17, -17], [-8, 8, -30, -8], [-24, 24, -6, -24], [7, -7, 35, 7], [-20, -14, 14, -14], [-15, -9, 9, -9], [-60, -4, 4, -4], [1, -1, -15, 1], [4, -4, 30, 4], [-14, 14, -12, -14], [18, -18, -6, 18], [6, -4, 4, -4], [8, 30, -30, 30], [0, 0, 46, 0], [-30, 4, -4, 4], [-24, -6, 6, -6], [10, -10, -48, 10], [-6, -22, 22, -22], [-32, 32, 6, -32], [10, 22, -22, 22], [-5, 5, -31, -5], [-10, 10, 14, -10], [-23, -17, 17, -17], [-20, 20, 16, -20], [-6, -32, 32, -32], [-2, 2, -16, -2], [-14, 12, -12, 12], [0, 0, 0, 0], [20, 14, -14, 14], [-7, 7, -39, -7], [-25, 21, -21, 21], [-16, 22, -22, 22], [-4, 22, -22, 22], [-18, -24, 24, -24], [27, -9, 9, -9], [2, -2, 44, 2], [-16, 16, -24, -16], [-1, 17, -17, 17], [-22, 8, -8, 8], [13, 25, -25, 25], [2, -2, -34, 2], [-16, 16, 4, -16], [30, 8, -8, 8], [6, 2, -2, 2], [-14, -12, 12, -12], [-19, 19, -1, -19], [16, -16, 2, 16], [-8, 8, 10, -8], [21, -9, 9, -9], [-6, -10, 10, -10], [-28, 28, 0, -28], [7, -13, 13, -13], [30, 12, -12, 12], [-30, 30, -12, -30], [18, -18, -28, 18], [26, -10, 10, -10], [-5, 5, 15, -5], [14, -14, 28, 14], [4, 6, -6, 6], [12, -12, -18, 12], [12, -6, 6, -6], [16, 12, -12, 12], [-11, -27, 27, -27], [-21, 21, 13, -21], [-25, 25, -19, -25], [22, -24, 24, -24], [-28, 28, -22, -28], [-24, 14, -14, 14], [9, 1, -1, 1], [2, 18, -18, 18], [-8, 8, 18, -8], [21, -21, 3, 21], [0, 0, -2, 0], [19, -19, 27, 19], [0, 0, 38, 0], [24, -14, 14, -14], [10, -24, 24, -24], [4, -4, -44, 4], [-26, 26, 16, -26], [-14, 14, 54, -14], [-24, -6, 6, -6], [4, -4, -70, 4], [20, -20, 26, 20], [-54, -4, 4, -4], [-21, 21, -5, -21], [-18, 18, -6, -18], [-36, 36, 0, -36], [5, -5, 73, 5], [-12, -10, 10, -10], [20, 0, 0, 0], [56, 16, -16, 16], [-29, 19, -19, 19], [21, -21, 33, 21], [30, -30, 20, 30], [10, 24, -24, 24], [-2, 2, -4, -2], [-40, -10, 10, -10], [-15, -25, 25, -25], [36, -6, 6, -6], [-6, 6, -4, -6], [27, -27, 23, 27], [12, 24, -24, 24], [6, 24, -24, 24], [26, -26, 12, 26], [41, -21, 21, -21], [42, 0, 0, 0], [16, -16, -24, 16], [0, 0, -36, 0], [50, 12, -12, 12], [-23, 23, -21, -23], [12, -12, -22, 12], [-32, 32, 6, -32], [-2, -14, 14, -14], [0, 0, -14, 0], [-4, 4, 42, -4], [-7, 7, -31, -7], [42, 0, 0, 0], [-20, 20, 30, -20], [36, 16, -16, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1560_w_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1560_2_w_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_1560_2_w_d(:prec:=4)
    chi := MakeCharacter_1560_w();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1560_2_w_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_1560_2_w_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1560_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![6, -6, 1]>,<11,R![36, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;