
// Make newform 2646.2.f.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2646_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2646_f_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_2646_2_f_m();".
// 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_2646_2_f_m();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 3, 3];
    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_2646_f();"
function MakeCharacter_2646_f()
    N := 2646;
    order := 3;
    char_gens := [785, 1081];
    v := [2, 3];
    // 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_2646_f_Hecke();"
function MakeCharacter_2646_f_Hecke(Kf)
    N := 2646;
    order := 3;
    char_gens := [785, 1081];
    char_values := [[-1, 0, 0, 0, 1, 0], [1, 0, 0, 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 := [[0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, 0], [-3, 0, -1, 0, 3, -2], [2, 2, 0, 0, 0, 0], [-2, -3, 0, 0, 0, 0], [2, 0, 0, 0, -2, -1], [0, -3, -2, -2, 0, 3], [-6, 0, -1, 0, 6, 1], [-1, 0, 0, 0, 0, 0], [1, 0, -2, 0, -1, 1], [0, 3, -3, -3, -2, -3], [0, 3, -3, -3, -3, -3], [5, -1, 0, 1, 0, 0], [-5, 0, 2, 0, 5, 1], [0, 1, -2, -2, -3, -1], [2, 0, -4, 0, -2, 1], [-4, -7, 0, 2, 0, 0], [8, 1, 0, 4, 0, 0], [0, -4, -1, -1, 0, 4], [0, -2, 3, 3, 1, 2], [2, -4, 0, 1, 0, 0], [0, 2, 2, 2, -8, -2], [0, -4, 3, 3, -10, 4], [-1, 0, 0, 0, 1, -3], [7, -5, 0, -1, 0, 0], [1, -1, 0, -1, 0, 0], [13, 0, -3, 0, -13, 1], [3, -5, 0, -5, 0, 0], [0, 0, -2, 0, 0, 0], [0, -5, -3, -3, -8, 5], [-5, 0, 4, 0, 5, -1], [1, 0, -8, 0, -1, -5], [0, 1, -2, -2, 2, -1], [-1, 0, -3, 0, 1, 3], [-7, -6, 0, -3, 0, 0], [5, 0, 0, 0, -5, -1], [0, -11, 1, 1, -2, 11], [-1, -7, 0, -2, 0, 0], [0, -1, 0, -4, 0, 0], [0, -6, -3, -3, -9, 6], [-5, 0, 3, 0, 5, 0], [11, 8, 0, -4, 0, 0], [13, 3, 0, 0, 0, 0], [1, 0, -6, 0, -1, -6], [0, 6, -6, -6, -7, -6], [0, 7, -1, -1, 13, -7], [-2, 0, 4, 0, 2, 2], [-6, -3, 0, -2, 0, 0], [5, 0, 2, 0, -5, -4], [0, 0, 0, 0, -13, 0], [-3, 6, 0, -3, 0, 0], [-4, 0, -1, 0, 4, 2], [0, 8, 1, 1, 2, -8], [0, -6, 0, -8, 0, 0], [3, -1, 0, 8, 0, 0], [0, 0, -9, -9, -14, 0], [0, -1, 9, 9, 5, 1], [-20, 0, -2, 0, 20, -1], [-5, 0, 10, 0, 5, 4], [8, -5, 0, 4, 0, 0], [1, 0, -1, 0, -1, -2], [0, 2, 2, 2, 4, -2], [0, -3, 9, 9, 3, 3], [0, -3, 3, 3, 16, 3], [-15, 0, -2, 0, 15, 2], [6, 0, -3, 0, -6, -6], [0, -1, 8, 8, -11, 1], [0, 1, -4, -4, -2, -1], [5, 5, 0, -9, 0, 0], [0, -6, 3, 3, -10, 6], [-4, 0, 5, 0, 4, 1], [13, 8, 0, -1, 0, 0], [10, 0, -3, 0, -10, -5], [0, -2, 3, 3, -23, 2], [-12, 2, 0, 8, 0, 0], [1, 0, 4, 0, -1, 13], [-4, 0, 3, 0, 4, 0], [10, 0, -2, 0, -10, -8], [0, 2, 2, 2, 0, -2], [11, 14, 0, -8, 0, 0], [4, 6, 0, -3, 0, 0], [0, -14, 1, 1, -9, 14], [0, 10, -5, -5, 4, -10], [-8, 4, 0, 7, 0, 0], [0, 1, -2, -2, 26, -1], [0, 3, -1, -1, 0, -3], [8, 0, 8, 0, -8, 13], [-11, 7, 0, -9, 0, 0], [0, -3, -8, -8, -3, 3], [-7, -15, 0, 0, 0, 0], [14, 0, 2, 0, -14, 5], [-1, 0, -13, 0, 1, -5], [-36, 0, 0, -3, 0, 0], [-1, 0, -4, 0, 1, -4], [1, -11, 0, -1, 0, 0], [-2, -12, 0, -9, 0, 0], [-15, 7, 0, 4, 0, 0], [0, -4, -7, -7, 12, 4], [18, -3, 0, 8, 0, 0], [20, 0, -5, 0, -20, 11], [0, 6, 0, 0, 21, -6], [3, 0, 7, 0, -3, 14], [26, -8, 0, 1, 0, 0], [0, -2, 9, 9, -8, 2], [-7, 14, 0, -8, 0, 0], [-15, 0, 3, 0, 15, 0], [0, 3, 3, 3, 17, -3], [-9, 0, 5, 0, 9, 1], [-6, 7, 0, -2, 0, 0], [-15, 0, -5, 0, 15, 15], [0, 11, -1, -1, -17, -11], [17, -3, 0, 15, 0, 0], [0, -3, -1, -1, 24, 3], [-13, 0, -6, 0, 13, -15], [-2, -5, 0, 5, 0, 0], [19, 0, 7, 0, -19, 13], [0, 5, 3, 3, 2, -5], [-8, 0, -5, 0, 8, -16], [0, 12, 12, 12, 7, -12], [0, -3, 0, 0, 0, 3], [-10, -7, 0, 4, 0, 0], [0, 9, 0, 0, 8, -9], [7, 19, 0, 4, 0, 0], [0, -9, 9, 9, -17, 9], [-31, -10, 0, 7, 0, 0], [0, -7, -1, -1, -17, 7], [-6, 0, -13, 0, 6, -5], [12, 16, 0, 1, 0, 0], [5, 0, 8, 0, -5, -4], [-6, 0, 1, 0, 6, 2], [-6, 4, 0, 1, 0, 0], [-5, 0, -3, 0, 5, -2], [-2, 0, -5, 0, 2, 20], [-18, -6, 0, -5, 0, 0], [-18, 0, 5, 0, 18, -11], [13, 0, -11, 0, -13, 13], [-3, 3, 0, 8, 0, 0], [5, -17, 0, -2, 0, 0], [0, 1, 7, 7, 31, -1], [-19, 0, -4, 0, 19, -11], [9, 9, 0, -6, 0, 0], [-19, 1, 0, -8, 0, 0], [0, 4, -3, -3, 34, -4], [0, 9, -3, -3, 14, -9], [0, -2, -13, -13, -5, 2], [11, 0, -9, 0, -11, 0], [-1, -1, 0, 0, 0, 0], [17, 0, 8, 0, -17, 4], [-21, -3, 0, -9, 0, 0], [-4, -6, 0, -12, 0, 0], [-1, 0, 11, 0, 1, -13], [0, -1, -16, -16, -3, 1], [0, -2, 2, 2, -29, 2], [13, 11, 0, -13, 0, 0], [0, 3, -3, -3, 42, -3], [7, 0, 0, 3, 0, 0], [-8, 0, 2, 0, 8, -2], [0, 8, -4, -4, -1, -8], [-4, -16, 0, 2, 0, 0], [37, 0, -3, 0, -37, -6], [5, -10, 0, 4, 0, 0], [-18, 0, -9, 0, 18, 6], [0, -11, 17, 17, 22, 11], [-22, -3, 0, 3, 0, 0], [0, 0, -3, -3, -4, 0], [-7, 9, 0, 3, 0, 0], [17, 0, 2, 0, -17, 14], [0, 1, 12, 12, -2, -1], [16, 0, -2, 0, -16, -7], [-29, 0, -8, 0, 29, -17], [0, -1, 5, 5, -3, 1], [18, 0, -13, 0, -18, 7], [-22, 0, -6, 0, 22, -10], [0, -10, 2, 2, -39, 10], [-41, -5, 0, 0, 0, 0], [28, 3, 0, 3, 0, 0], [0, -21, 6, 6, -13, 21], [0, 8, -13, -13, -15, -8], [0, -11, 3, 3, 1, 11], [8, 0, 11, 0, -8, -14], [-20, 7, 0, -18, 0, 0], [14, 0, 10, 0, -14, -10], [0, 13, 11, 11, -26, -13], [-3, -26, 0, -8, 0, 0], [0, 9, 6, 6, 32, -9], [-39, 0, 10, 0, 39, 5], [-29, 4, 0, 8, 0, 0], [-16, -14, 0, -5, 0, 0], [0, -2, -10, -10, -2, 2], [-17, -1, 0, -13, 0, 0], [0, -4, 2, 2, -25, 4], [-13, -10, 0, 4, 0, 0], [-10, 0, 22, 0, 10, 17], [1, 0, 15, 0, -1, 15], [0, 22, -17, -17, 2, -22], [0, -10, -11, -11, -1, 10], [-10, -7, 0, -2, 0, 0], [-6, 0, 15, 0, 6, -9], [0, 4, -11, -11, 15, -4], [-34, 0, 0, 0, 34, 9], [0, -20, 4, 4, 9, 20], [-12, 15, 0, 0, 0, 0], [19, 16, 0, -4, 0, 0], [4, -6, 0, 0, 0, 0], [13, 0, 7, 0, -13, 16], [0, -17, 5, 5, -26, 17], [26, 0, 6, 0, -26, 12], [2, -24, 0, -6, 0, 0], [20, 0, -3, 0, -20, -13], [0, 14, -16, -16, 3, -14], [0, 13, -8, -8, 4, -13], [-3, 0, 6, 0, 3, -9], [0, -15, -12, -12, 2, 15], [32, 0, -1, 0, -32, 4], [0, 18, -14, -14, -9, -18], [-9, -15, 0, -11, 0, 0], [-27, 0, 7, 0, 27, -15], [-10, 0, 8, 0, 10, -14], [37, 0, -2, 0, -37, 14], [7, 0, -20, 0, -7, -8], [31, -7, 0, -1, 0, 0], [-23, 0, 7, 0, 23, 13], [0, 24, 9, 9, 19, -24], [0, 8, 20, 20, 14, -8], [-3, 21, 0, -10, 0, 0], [0, 16, -8, -8, 3, -16], [0, -12, 9, 9, -33, 12], [18, 0, -8, 0, -18, -7], [-34, 13, 0, 7, 0, 0], [-31, 0, -1, 0, 31, -17], [30, 0, -21, 0, -30, 0], [0, -6, 6, 6, 8, 6], [0, -2, 20, 20, 31, 2], [-5, 0, 4, 0, 5, -1], [-38, 4, 0, -19, 0, 0], [35, 0, -13, 0, -35, -10], [5, -25, 0, -13, 0, 0], [0, 18, -10, -10, 0, -18], [-29, 6, 0, -6, 0, 0], [-25, 0, 3, 0, 25, 6], [15, 1, 0, 1, 0, 0], [-12, 0, -4, 0, 12, 3], [0, -15, -12, -12, -21, 15], [-21, 14, 0, -19, 0, 0], [-22, 0, -14, 0, 22, -4], [10, 0, 3, 0, -10, 6], [-24, 3, 0, -21, 0, 0], [11, 0, -10, 0, -11, -20], [14, -1, 0, -11, 0, 0], [18, 0, 3, 0, -18, 24], [0, -7, 14, 14, 28, 7], [0, 12, -8, -8, -24, -12], [3, -21, 0, -5, 0, 0], [-16, 3, 0, -9, 0, 0], [0, -14, -14, -14, 6, 14], [-31, -16, 0, 11, 0, 0], [13, 3, 0, -9, 0, 0], [0, -8, 13, 13, 11, 8], [0, 3, 13, 13, 24, -3], [25, 0, 4, 0, -25, -10], [-28, 4, 0, -14, 0, 0], [-24, 0, -15, 0, 24, -12], [0, 17, -13, -13, 19, -17], [12, 3, 0, -15, 0, 0], [0, -9, 2, 2, -66, 9], [32, 0, -10, 0, -32, 10], [-33, 0, 27, 0, 33, 9], [39, 0, 11, 0, -39, 22], [26, 18, 0, -15, 0, 0], [0, 4, -5, -5, -12, -4], [-2, 0, 3, 0, 2, 9], [22, 0, -8, 0, -22, 8], [15, 2, 0, -7, 0, 0], [38, 0, -6, 0, -38, 17], [0, 8, 17, 17, -3, -8], [22, 38, 0, -10, 0, 0], [0, -18, 3, 3, -24, 18], [24, 0, -12, 0, -24, 9], [-27, 0, -2, 0, 27, -7], [0, -10, -2, -2, 53, 10], [0, 9, -9, -9, -31, -9], [-9, 0, 29, 0, 9, 1], [27, -15, 0, 13, 0, 0], [6, 19, 0, 1, 0, 0], [24, 0, -5, 0, -24, 18], [0, 0, 3, 3, 23, 0], [36, 0, 0, -12, 0, 0], [0, 14, 16, 16, -19, -14], [-26, -38, 0, 8, 0, 0], [-41, 0, 7, 0, 41, -11], [2, 0, 18, 0, -2, 5], [0, -9, 15, 15, -23, 9], [9, 0, 21, 0, -9, -24], [0, 9, -9, -9, -10, -9], [0, 2, 16, 16, 2, -2], [44, -7, 0, -8, 0, 0], [0, 4, -2, -2, 51, -4], [7, 0, 4, 0, -7, -25], [1, 7, 0, -13, 0, 0], [10, -1, 0, 23, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2646_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_f_m();".
// 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_2646_2_f_m(:prec:=6)
    chi := MakeCharacter_2646_f();
    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_2646_2_f_m();".
// 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_2646_2_f_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2646_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, 18, 33, 12, 7, -1, 1]>,<11,R![9, 18, 33, 12, 7, -1, 1]>,<13,R![4761, -69, 553, 146, 63, 8, 1]>],Snew);
    return Vf;
end function;