
// Make newform 1960.2.q.p in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1960_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1960_q_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_1960_2_q_p();".
// 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_1960_2_q_p();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_1960_q();"
function MakeCharacter_1960_q()
    N := 1960;
    order := 3;
    char_gens := [1471, 981, 1177, 1081];
    v := [3, 3, 3, 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_1960_q_Hecke();"
function MakeCharacter_1960_q_Hecke(Kf)
    N := 1960;
    order := 3;
    char_gens := [1471, 981, 1177, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 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 := 2;
    raw_aps := [[0, 0, 0, 0], [-1, 1, -1, 0], [0, 0, -1, 0], [0, 0, 0, 0], [1, 0, 1, 0], [1, 0, 0, -1], [-1, 1, -1, 0], [0, 0, 2, 0], [0, -3, 2, -3], [1, 0, 0, 0], [-6, 3, -6, 0], [0, -7, 2, -7], [6, 0, 0, -1], [-6, 0, 0, -4], [0, 1, 9, 1], [-8, -1, -8, 0], [-6, -3, -6, 0], [0, 2, -4, 2], [0, -1, 0, 0], [6, 0, 0, -6], [8, -2, 8, 0], [0, -4, -9, -4], [0, 0, 0, -8], [0, -4, 4, -4], [7, 0, 0, 5], [-2, 6, -2, 0], [0, 3, 11, 3], [0, 6, -6, 6], [7, -8, 7, 0], [-2, 0, 0, -5], [8, 0, 0, -3], [0, -3, 16, -3], [8, -8, 8, 0], [0, 0, 0, 1], [0, -2, -12, -2], [1, -4, 1, 0], [-2, 6, -2, 0], [0, 5, 10, 5], [9, 0, 0, -1], [0, 9, -1, 9], [10, 10, 10, 0], [-10, 0, 0, -11], [0, -2, 11, -2], [-12, -8, -12, 0], [-8, 0, 0, -3], [-6, -13, -6, 0], [3, 0, 0, 10], [1, 0, 0, -13], [-3, -1, -3, 0], [0, -7, -14, -7], [0, 6, 12, 6], [1, 0, 0, -4], [8, 13, 8, 0], [-8, 0, 0, 11], [0, -6, 2, -6], [-2, 12, -2, 0], [-20, -1, -20, 0], [0, 4, 8, 4], [6, -5, 6, 0], [7, 0, 0, 8], [-1, 17, -1, 0], [-17, 0, 0, -1], [-19, 0, 0, 5], [-10, 0, -10, 0], [0, 9, -11, 9], [0, 4, 18, 4], [0, 6, 18, 6], [-6, 0, 0, -5], [-2, 19, -2, 0], [-18, 0, 0, 8], [-9, 15, -9, 0], [0, 4, -24, 4], [-7, 3, -7, 0], [0, -2, 20, -2], [6, 0, 0, 16], [0, -2, 20, -2], [9, 14, 9, 0], [0, -1, -7, -1], [0, 18, 1, 18], [2, -22, 2, 0], [-30, 0, 0, 1], [-15, 0, 0, -8], [-17, 4, -17, 0], [-10, 0, 0, 12], [0, -3, -8, -3], [0, 26, 2, 26], [-19, 0, 0, -6], [0, 5, 12, 5], [-4, 0, 0, -20], [0, 0, 0, 18], [0, -1, 7, -1], [8, -23, 8, 0], [-14, -9, -14, 0], [-3, 0, 0, -6], [0, 14, -7, 14], [7, 0, 0, -17], [0, -3, 12, -3], [-6, 0, -6, 0], [0, -6, -24, -6], [0, 8, -1, 8], [4, 0, 0, 22], [12, 10, 12, 0], [6, 24, 6, 0], [0, 10, -20, 10], [-2, -18, -2, 0], [35, -1, 35, 0], [-28, 0, 0, 10], [0, 11, 25, 11], [25, -8, 25, 0], [10, 0, 0, -20], [0, -5, 13, -5], [-18, 8, -18, 0], [-10, 0, 0, -1], [8, -5, 8, 0], [-15, 0, 0, -16], [-4, 20, -4, 0], [-9, 0, 0, -15], [-4, 10, -4, 0], [0, 20, 6, 20], [9, 0, 0, 6], [28, -2, 28, 0], [48, 0, 0, 2], [0, -5, 29, -5], [16, 2, 16, 0], [0, -10, -4, -10], [-23, 0, 0, -10], [0, -4, -27, -4], [0, -21, 20, -21], [30, 0, 0, 8], [0, 21, 3, 21], [-15, -14, -15, 0], [4, 0, 0, -7], [0, -16, 3, -16], [0, 0, 0, -26], [0, 11, 20, 11], [-10, 0, 0, -6], [3, -13, 3, 0], [-1, -31, -1, 0], [43, 0, 0, -5], [-15, 24, -15, 0], [-24, 0, 0, -11], [0, -6, 13, -6], [12, -17, 12, 0], [-30, 0, 0, -17], [-6, 1, -6, 0], [12, 0, 0, -3], [-2, 0, 0, 8], [-46, 6, -46, 0], [0, 18, -28, 18], [0, -27, -16, -27], [0, -12, 10, -12], [12, 0, 0, 4], [4, 0, 0, 22], [0, 20, 10, 20], [-22, -21, -22, 0], [32, 0, 0, -8], [0, -28, -11, -28], [0, 23, -2, 23], [-7, 0, 0, 15], [-24, 0, -24, 0], [0, -7, 0, -7], [40, 0, 0, 13], [26, 0, 0, 2], [0, 27, 10, 27], [6, 34, 6, 0], [-15, 13, -15, 0], [-6, -8, -6, 0], [3, -7, 3, 0], [-1, 0, 0, -40], [0, 6, 4, 6], [-26, 20, -26, 0], [22, 0, 0, -26], [0, -10, -27, -10], [-10, -13, -10, 0], [-34, -8, -34, 0], [12, 0, 0, -22], [13, 0, 0, 28], [35, 0, 35, 0], [42, 0, 0, 0], [0, -10, 16, -10], [0, -12, 36, -12], [-4, 0, 0, 14], [2, 0, 0, -34], [0, -12, 2, -12], [0, 12, 0, 0], [4, 17, 4, 0], [-12, 3, -12, 0], [19, 3, 19, 0], [0, -2, 13, -2], [12, -18, 12, 0], [0, 8, -10, 8], [-26, 0, 0, -6], [0, -2, -17, -2], [0, -13, 50, -13], [44, -6, 44, 0], [-39, -13, -39, 0], [49, 6, 49, 0], [0, 1, -16, 1], [15, 0, 0, -33], [0, -23, -21, -23], [16, -30, 16, 0], [6, 0, 0, -14], [0, 20, -6, 20], [42, -6, 42, 0], [10, 0, 0, -10], [-9, -11, -9, 0], [0, 17, 16, 17], [0, 6, -8, 6], [-28, 0, 0, 12], [-28, 8, -28, 0], [0, -21, -4, -21], [8, 0, 0, 19], [-32, 0, 0, -3], [0, -15, 11, -15], [18, -20, 18, 0], [0, -10, -28, -10], [-42, -1, -42, 0], [14, -3, 14, 0], [0, -2, 38, -2], [-30, 0, 0, 7], [0, 8, -26, 8], [6, 0, 0, 28], [0, -2, 1, -2], [0, 16, -30, 16], [11, 0, 0, 31], [34, 0, 0, -4], [0, 13, -3, 13], [-17, 18, -17, 0], [0, 17, 27, 17], [0, 28, -10, 28], [4, -22, 4, 0], [30, 28, 30, 0], [10, 0, 0, 38], [-1, -8, -1, 0], [-1, 0, 0, 5], [5, 35, 5, 0], [0, 7, 10, 7], [0, 2, -4, 2], [21, 0, 0, -12], [0, 0, 0, 21], [-4, 44, -4, 0], [0, -4, -48, -4], [-27, -33, -27, 0], [0, 14, 16, 14], [-17, 0, 0, 25], [0, 8, -2, 8], [-4, 0, 0, 20], [-26, -2, -26, 0], [-5, -34, -5, 0], [-14, 0, 0, -18], [-18, 0, 0, 32], [0, -12, -58, -12], [4, -1, 4, 0], [12, 0, 0, 8], [6, -30, 6, 0], [0, 14, 39, 14], [72, 2, 72, 0], [-50, -16, -50, 0], [-33, 0, 0, 9], [0, 11, 13, 11], [-2, 34, -2, 0], [14, 0, 0, 5], [10, 45, 10, 0], [-69, 0, 0, 5], [62, -4, 62, 0], [0, 29, -16, 29], [-3, 0, 0, 34], [26, 0, 0, 34], [0, 0, 0, 23], [2, 20, 2, 0], [0, -33, -14, -33], [32, 3, 32, 0], [5, -17, 5, 0], [0, 36, 2, 36], [30, 0, 0, 28], [0, 13, -23, 13], [0, -2, -48, -2], [-52, 16, -52, 0], [0, -8, 7, -8], [0, 10, -34, 10], [-15, -37, -15, 0], [-7, 38, -7, 0], [-20, 0, 0, -12], [-44, 0, 0, -19], [0, -31, 1, -31], [0, -8, 5, -8], [2, 42, 2, 0], [18, 0, 0, -4], [-22, -7, -22, 0], [-50, 0, 0, -9], [-27, -10, -27, 0], [-25, -13, -25, 0], [0, 7, -28, 7], [6, 0, 0, 5], [-54, 0, 0, 4], [-18, 47, -18, 0], [0, -7, 56, -7], [-5, 0, 0, -29], [0, 30, 0, 30], [-33, 0, 0, 21], [0, -31, -45, -31], [0, 7, 14, 7], [19, 12, 19, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1960_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1960_2_q_p();".
// 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_1960_2_q_p(:prec:=4)
    chi := MakeCharacter_1960_q();
    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_1960_2_q_p();".
// 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_1960_2_q_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1960_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -2, 5, 2, 1]>,<11,R![1, -1, 1]>,<13,R![-1, -2, 1]>],Snew);
    return Vf;
end function;