
// Make newform 1960.2.q.u in Magma, downloaded from the LMFDB on 25 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_u();".
// 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_u();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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], [-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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, 1, 0, 0], [1, 0, -1, 0], [0, 0, 0, 0], [0, 1, 0, 0], [-2, 0, 0, -3], [0, -1, 6, 0], [-4, -2, 4, -2], [0, -2, 0, -2], [2, 0, 0, -1], [0, -4, 0, 0], [2, 4, -2, 4], [-2, 0, 0, 2], [-4, 0, 0, -2], [4, -1, -4, -1], [0, -2, 10, 0], [0, 0, -4, 0], [-10, 2, 10, 2], [0, 4, 4, 0], [0, 0, 0, 0], [0, 4, 2, 0], [4, -3, -4, -3], [-12, 0, 0, 0], [2, 2, -2, 2], [-6, 0, 0, 3], [0, 0, 6, 0], [-4, -5, 4, -5], [-12, -2, 12, -2], [0, 1, 6, 0], [2, 0, 0, 4], [0, 0, 0, -4], [12, 2, -12, 2], [0, -6, -2, 0], [12, 0, 0, 6], [-6, 4, 6, 4], [0, -5, 12, 0], [0, -4, -2, 0], [-4, -10, 4, -10], [-12, 0, 0, 5], [-6, -11, 6, -11], [0, 0, 4, 0], [2, 0, 0, 2], [-4, 3, 4, 3], [0, 2, -10, 0], [-10, 0, 0, -2], [0, 4, -8, 0], [-16, 0, 0, -1], [-20, 0, 0, -3], [0, -3, -16, 0], [-2, -6, 2, -6], [14, -2, -14, -2], [-4, 0, 0, -9], [0, 0, 2, 0], [-4, 0, 0, -6], [-6, -8, 6, -8], [0, 6, 0, 0], [0, -4, -2, 0], [0, -4, 0, -4], [0, -4, 2, 0], [-18, 0, 0, -1], [0, 3, 0, 0], [-18, 0, 0, -9], [24, 0, 0, 3], [0, -2, -16, 0], [22, 3, -22, 3], [10, 0, -10, 0], [12, 0, -12, 0], [2, 0, 0, 0], [0, 2, -12, 0], [18, 0, 0, -4], [0, -5, -10, 0], [-32, 0, 32, 0], [0, -7, 4, 0], [-6, -6, 6, -6], [20, 0, 0, 8], [-16, 4, 16, 4], [0, 13, -2, 0], [-6, -11, 6, -11], [2, 7, -2, 7], [0, 8, -6, 0], [12, 0, 0, 4], [2, 0, 0, 1], [0, -7, 12, 0], [-26, 0, 0, 4], [16, 2, -16, 2], [4, 10, -4, 10], [-26, 0, 0, 3], [-18, 6, 18, 6], [-22, 0, 0, 2], [0, 0, 0, -4], [24, 1, -24, 1], [0, -6, 8, 0], [0, 2, 0, 0], [-8, 0, 0, 3], [8, -5, -8, -5], [12, 0, 0, 11], [14, 12, -14, 12], [0, 4, -6, 0], [-12, 0, 12, 0], [-18, -15, 18, -15], [12, 0, 0, 0], [0, -10, 2, 0], [0, -8, -20, 0], [6, -16, -6, -16], [0, 0, -20, 0], [0, 1, -2, 0], [-4, 0, 0, -4], [14, 15, -14, 15], [0, -7, 36, 0], [22, 0, 0, 0], [-4, -3, 4, -3], [0, 12, 2, 0], [-22, 0, 0, 4], [0, 6, 20, 0], [-4, 0, 0, -15], [0, -16, -2, 0], [16, 0, 0, 9], [0, 0, 8, 0], [26, 8, -26, 8], [-8, 0, 0, 5], [0, 10, -2, 0], [26, 0, 0, 10], [-6, 3, 6, 3], [0, 16, -20, 0], [4, 8, -4, 8], [-14, 0, 0, 3], [-10, 9, 10, 9], [-8, -10, 8, -10], [-24, 0, 0, -16], [18, -1, -18, -1], [0, 1, 8, 0], [-24, 0, 0, 0], [36, -3, -36, -3], [6, 0, 0, 6], [26, 0, -26, 0], [-18, 0, 0, -16], [0, 5, -22, 0], [0, -11, 16, 0], [-10, 0, 0, -5], [0, 9, 10, 0], [12, 0, 0, 10], [-34, -13, 34, -13], [0, 4, 24, 0], [28, 0, 0, -2], [0, -10, -10, 0], [-16, 0, 0, -2], [-22, 0, 0, 4], [0, -12, 34, 0], [12, 16, -12, 16], [-16, -8, 16, -8], [26, -8, -26, -8], [-26, 0, 0, -18], [28, 0, 0, -4], [-24, -8, 24, -8], [0, -18, 20, 0], [8, 0, 0, 0], [-4, 1, 4, 1], [-6, 2, 6, 2], [-30, 0, 0, -5], [0, 6, 38, 0], [4, 20, -4, 20], [-30, 0, 0, -14], [16, 0, 0, 18], [-4, -12, 4, -12], [0, 16, 14, 0], [0, 11, -12, 0], [0, -8, 8, 0], [0, 7, -46, 0], [14, 0, 0, -11], [38, -8, -38, -8], [0, -4, -36, 0], [34, 0, 0, 8], [36, 11, -36, 11], [0, 4, 22, 0], [0, -2, -24, 0], [-18, 0, 0, -14], [16, 0, 0, 21], [0, -19, 30, 0], [-8, 0, 0, -12], [6, 18, -6, 18], [32, 0, -32, 0], [-36, 0, 0, -12], [-2, 0, 0, -4], [-14, -8, 14, -8], [0, -8, 0, 0], [0, 0, 22, 0], [0, -6, 34, 0], [0, 1, 64, 0], [10, -1, -10, -1], [0, -4, 16, 0], [-22, 12, 22, 12], [-20, 0, 0, -2], [-16, 5, 16, 5], [6, -2, -6, -2], [0, -16, -12, 0], [0, 1, 6, 0], [0, -13, 10, 0], [18, -18, -18, -18], [26, 0, 0, -1], [28, 19, -28, 19], [0, 8, -30, 0], [8, 0, 0, 10], [-42, 0, 42, 0], [0, 8, 18, 0], [-52, 0, 0, 6], [0, -17, 10, 0], [8, 2, -8, 2], [-44, -4, 44, -4], [-6, 0, 0, -28], [0, 0, 12, 0], [-26, -22, 26, -22], [-6, 0, 0, 12], [24, 0, 0, 4], [8, 27, -8, 27], [0, 2, 48, 0], [-14, 10, 14, 10], [0, -22, 24, 0], [0, -22, 26, 0], [0, 26, 0, 26], [6, 0, 0, 28], [-6, 12, 6, 12], [0, 0, 0, 26], [-30, -21, 30, -21], [-24, -10, 24, -10], [0, 0, 0, 5], [22, 0, 0, -8], [46, 5, -46, 5], [0, 1, 44, 0], [44, -5, -44, -5], [-4, -16, 4, -16], [0, -10, 18, 0], [0, -6, 12, 0], [-8, 0, 0, 8], [0, 1, -22, 0], [40, 0, 0, 5], [0, 5, -28, 0], [10, 26, -10, 26], [-22, -30, 22, -30], [-56, 0, 0, 3], [16, 0, 0, 10], [0, -4, -28, 0], [-20, 20, 20, 20], [0, 9, -20, 0], [2, 16, -2, 16], [18, 0, 0, 9], [0, -10, 0, -10], [-32, 0, 0, 0], [0, -2, -36, 0], [0, 7, -8, 0], [24, 0, 0, -6], [54, 0, 0, 0], [18, 4, -18, 4], [0, -16, -24, 0], [-2, 0, 0, 14], [0, -4, -2, 0], [-8, 11, 8, 11], [0, 0, -22, 0], [0, 16, -28, 0], [22, 0, 0, 5], [6, 15, -6, 15], [0, 2, -8, 0], [-12, 0, 0, 10], [0, 18, -18, 0], [14, 0, 0, 1], [0, 12, 10, 0], [-36, -22, 36, -22], [-6, 0, 0, -3], [-10, 0, 0, 4], [-44, 0, 0, -4], [0, -24, 2, 0], [-10, 14, 10, 14], [0, 0, 4, 0], [0, 3, -18, 0], [40, 4, -40, 4], [22, 0, 0, 8], [12, 15, -12, 15], [-36, 8, 36, 8], [0, 12, -30, 0], [-22, 15, 22, 15], [44, 2, -44, 2], [0, -11, -12, 0], [0, 3, -36, 0], [-8, 0, 0, 8], [-6, 0, 0, -8], [40, -13, -40, -13], [-4, 31, 4, 31], [0, 0, 30, 0], [-2, 0, 0, -8], [0, -24, -24, 0], [38, 0, 0, -18], [0, 13, 6, 0], [0, -1, 8, 0], [30, 22, -30, 22], [36, 0, 0, 16], [38, 0, 0, 0], [0, 2, 38, 0], [-24, -22, 24, -22], [54, 0, 0, -3], [-20, -16, 20, -16], [-32, 0, 0, -15], [30, 1, -30, 1], [-22, -12, 22, -12], [0, 5, 12, 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_u();".
// 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_u(: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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1960_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 4, 5, -1, 1]>,<11,R![16, 4, 5, -1, 1]>,<13,R![-38, 1, 1]>],Snew);
    return Vf;
end function;