
// Make newform 2646.2.e.o in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2646_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2646_e_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_e_o();".
// 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_e_o();".
// 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_e();"
function MakeCharacter_2646_e()
    N := 2646;
    order := 3;
    char_gens := [785, 1081];
    v := [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_2646_e_Hecke();"
function MakeCharacter_2646_e_Hecke(Kf)
    N := 2646;
    order := 3;
    char_gens := [785, 1081];
    char_values := [[0, 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 := [[-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -1], [0, 0, 0, 0, 0, 0], [0, 1, -2, -2, 0, -1], [0, 1, 0, 0, 1, -1], [0, 0, -2, 0, 0, 2], [0, -1, 1, 1, 1, 1], [3, 0, -1, 0, -3, 1], [1, 0, -1, 0, -1, -3], [-5, -2, 0, 1, 0, 0], [0, 2, -2, -2, -3, -2], [0, -1, 1, 1, -4, 1], [5, 0, 1, 0, -5, -2], [-2, -2, 0, -1, 0, 0], [-2, 0, 1, 0, 2, 4], [0, 5, 0, -1, 0, 0], [2, 5, 0, -3, 0, 0], [-3, -1, 0, -3, 0, 0], [-1, 1, 0, 3, 0, 0], [9, 0, -3, 0, -9, -1], [0, 3, 0, 4, 0, 0], [3, 0, -2, 0, -3, -1], [0, -1, -2, -2, -4, 1], [10, 0, -2, 0, -10, 0], [0, 3, -4, -4, -4, -3], [2, 0, -3, 0, -2, -3], [0, 0, 3, 3, 6, 0], [-2, 0, 1, 0, 2, 2], [0, 4, 3, 3, 2, -4], [-4, -4, 0, 5, 0, 0], [-4, 0, 4, 0, 4, 6], [0, 0, 1, 1, -5, 0], [0, 3, -1, -1, 10, -3], [0, 0, 3, 0, 0, 3], [0, 5, -1, -1, 13, -5], [-8, 6, 0, 3, 0, 0], [0, 7, 2, 2, -7, -7], [0, 7, 1, 1, 6, -7], [11, 4, 0, 3, 0, 0], [0, 0, -3, 0, 0, -3], [-17, 1, 0, -5, 0, 0], [11, 5, 0, -2, 0, 0], [-9, -5, 0, 2, 0, 0], [-9, 2, 0, 2, 0, 0], [6, 0, 1, 0, -6, 1], [0, 4, -4, -4, 11, -4], [7, 0, 2, 0, -7, 2], [0, -8, 5, 5, -8, 8], [4, 0, -2, 0, -4, 6], [0, 1, -5, -5, -9, -1], [0, -8, -4, -4, 1, 8], [0, -4, -8, -8, -5, 4], [-17, 1, 0, 2, 0, 0], [4, 0, -8, 0, -4, -5], [0, -1, 4, 4, -10, 1], [6, 0, -10, 0, -6, -2], [0, 9, -5, -5, 0, -9], [0, 3, 0, 0, 4, -3], [0, 0, -1, 0, 0, -8], [-7, 3, 0, -5, 0, 0], [0, 8, -2, -2, 11, -8], [-1, 3, 0, 1, 0, 0], [-3, -7, 0, 2, 0, 0], [10, 8, 0, 2, 0, 0], [-6, -6, 0, 9, 0, 0], [11, -6, 0, 3, 0, 0], [0, -8, -2, -2, -17, 8], [16, 1, 0, 8, 0, 0], [0, 0, -9, 0, 0, -1], [0, 9, -3, -3, -3, -9], [0, -2, 9, 9, -4, 2], [0, -1, -2, -2, 28, 1], [-21, 0, 3, 0, 21, -2], [-3, 5, 0, -12, 0, 0], [11, 0, 3, 0, -11, -2], [0, 3, -8, -8, -11, -3], [0, 2, 2, 2, 14, -2], [4, 0, -5, 0, -4, -11], [-10, -5, 0, 2, 0, 0], [0, -10, 4, 4, 2, 10], [14, 0, -6, 0, -14, 8], [-1, 0, 2, 0, 1, 8], [-4, -5, 0, 8, 0, 0], [11, 5, 0, -6, 0, 0], [14, 1, 0, 0, 0, 0], [-2, -13, 0, -6, 0, 0], [19, 9, 0, -1, 0, 0], [-3, 0, 5, 0, 3, 4], [0, -13, -1, -1, 7, 13], [0, 4, 7, 7, 12, -4], [0, -3, -5, -5, 16, 3], [-8, 0, 3, 0, 8, -9], [0, -5, -13, -13, -5, 5], [10, 0, 7, 0, -10, 2], [8, 11, 0, -8, 0, 0], [-5, 0, -2, 0, 5, -14], [-18, 0, 15, 0, 18, 0], [0, -9, 6, 6, -10, 9], [0, -7, -9, -9, -16, 7], [12, 0, -2, 0, -12, 15], [1, 0, -7, 0, -1, -15], [-11, -10, 0, -3, 0, 0], [-3, 6, 0, 6, 0, 0], [-9, -9, 0, -2, 0, 0], [-12, 0, 6, 0, 12, 11], [14, 0, 0, 0, -14, -11], [0, -6, -6, -6, 21, 6], [-29, -5, 0, -4, 0, 0], [8, 0, 3, 0, -8, 6], [14, 0, 1, 0, -14, -10], [-5, 0, -1, 0, 5, -11], [0, -8, 1, 1, 0, 8], [0, -4, -3, -3, -6, 4], [-20, -10, 0, -5, 0, 0], [0, -12, -1, -1, -7, 12], [0, -9, -3, -3, 8, 9], [33, 0, -3, 0, -33, 0], [22, 0, 1, 0, -22, 10], [-9, 0, 11, 0, 9, -2], [2, 3, 0, -8, 0, 0], [-9, 0, -4, 0, 9, -4], [-9, 9, 0, 3, 0, 0], [-11, 0, -5, 0, 11, 7], [-5, -9, 0, -3, 0, 0], [30, 7, 0, 1, 0, 0], [-8, -4, 0, -5, 0, 0], [0, -1, -2, -2, 17, 1], [-14, 0, 1, 0, 14, -18], [-17, 0, -3, 0, 17, -14], [8, 0, -6, 0, -8, 17], [0, -6, 0, 0, 9, 6], [-36, 0, -2, 0, 36, -9], [8, 17, 0, 8, 0, 0], [15, 0, 4, 0, -15, 17], [0, 13, -2, -2, 8, -13], [22, 0, -6, 0, -22, 11], [13, -2, 0, -5, 0, 0], [0, -4, -5, -5, -16, 4], [-14, 0, -1, 0, 14, 9], [4, -7, 0, -13, 0, 0], [-10, 4, 0, -3, 0, 0], [8, 7, 0, 13, 0, 0], [-2, 7, 0, -7, 0, 0], [2, 0, -13, 0, -2, -5], [10, 0, 8, 0, -10, 3], [21, 0, 7, 0, -21, 4], [0, 5, 4, 4, -1, -5], [23, 0, 0, 0, -23, 15], [0, -11, 3, 3, 8, 11], [9, 0, 6, 0, -9, -2], [32, 8, 0, -5, 0, 0], [-14, -10, 0, -14, 0, 0], [32, 0, -1, 0, -32, 2], [0, 7, 7, 7, 16, -7], [5, 0, 2, 0, -5, 14], [0, -4, 11, 11, 10, 4], [11, -4, 0, -5, 0, 0], [-7, 7, 0, 14, 0, 0], [-10, 4, 0, 6, 0, 0], [5, -12, 0, -10, 0, 0], [0, -10, 0, 8, 0, 0], [0, 1, -10, -10, -1, -1], [0, -2, -4, -4, 13, 2], [-10, -7, 0, 4, 0, 0], [0, -14, -1, -1, 13, 14], [-1, 0, -5, 0, 1, -8], [0, 13, 8, 8, -30, -13], [10, -4, 0, 19, 0, 0], [7, 0, -8, 0, -7, 10], [0, 13, -9, -9, -13, -13], [0, -11, 13, 13, 17, 11], [0, 0, -9, 0, 0, 15], [0, 16, -11, -11, -8, -16], [17, -4, 0, 15, 0, 0], [0, -8, 22, 22, 24, 8], [-1, 0, 12, 0, 1, 20], [-2, 0, 21, 0, 2, 5], [-7, -14, 0, -7, 0, 0], [-14, 15, 0, 3, 0, 0], [7, 13, 0, 0, 0, 0], [-26, 0, -7, 0, 26, 6], [0, -9, 10, 10, -28, 9], [0, -1, -3, -3, -29, 1], [48, 4, 0, -2, 0, 0], [0, 2, 13, 13, 5, -2], [21, 0, -8, 0, -21, 0], [0, 3, 15, 15, 17, -3], [-8, 0, -13, 0, 8, -5], [-3, 0, 0, 0, 3, 6], [0, -5, -20, -20, -4, 5], [0, 0, -18, 0, 0, -12], [0, -16, -1, -1, 10, 16], [-19, 12, 0, 14, 0, 0], [-9, 0, 0, 0, 9, -30], [11, 9, 0, 11, 0, 0], [12, -8, 0, -1, 0, 0], [-18, -1, 0, 6, 0, 0], [-9, 0, -12, 0, 9, -9], [0, -5, 5, 5, 19, 5], [17, -10, 0, 13, 0, 0], [-1, 0, -8, 0, 1, -7], [-5, 0, 11, 0, 5, 11], [0, 12, -8, -8, -27, -12], [-1, -25, 0, 1, 0, 0], [-19, 0, -10, 0, 19, -22], [0, -2, -10, -10, -18, 2], [-35, 0, 4, 0, 35, -5], [5, 0, 11, 0, -5, -16], [30, -10, 0, 28, 0, 0], [0, -6, -12, -12, 10, 6], [0, 0, -13, -13, -43, 0], [5, 0, -6, 0, -5, -16], [7, 3, 0, 1, 0, 0], [-34, -1, 0, -5, 0, 0], [7, 3, 0, -9, 0, 0], [-8, -15, 0, 8, 0, 0], [0, 4, -18, -18, -43, -4], [0, -8, -11, -11, -18, 8], [0, 10, -5, -5, -12, -10], [-8, 0, 10, 0, 8, 14], [0, -2, 10, 10, 11, 2], [-33, 0, 0, 0, 33, -12], [0, -2, -15, -15, -18, 2], [0, 8, -17, -17, 14, -8], [-35, 0, 6, 0, 35, 3], [14, 0, 0, -16, 0, 0], [-28, 0, -9, 0, 28, -11], [9, 18, 0, -4, 0, 0], [-3, 9, 0, -12, 0, 0], [-23, 1, 0, 15, 0, 0], [-7, 0, -19, 0, 7, -2], [0, -18, 13, 13, -22, 18], [4, -5, 0, -22, 0, 0], [-2, 0, 2, 0, 2, 20], [0, 28, 2, 2, 25, -28], [0, 0, -15, 0, 0, -1], [0, -9, -12, -12, 24, 9], [0, -3, 6, 6, 21, 3], [-18, 8, 0, -7, 0, 0], [0, -24, 2, 2, -10, 24], [0, -4, -8, -8, 19, 4], [7, -3, 0, -18, 0, 0], [0, -6, -7, -7, 30, 6], [0, 7, -9, -9, 29, -7], [-34, -13, 0, 1, 0, 0], [5, -13, 0, 18, 0, 0], [6, -2, 0, 4, 0, 0], [8, 15, 0, 0, 0, 0], [21, 30, 0, -15, 0, 0], [0, 32, 2, 2, 7, -32], [0, -8, -25, -25, -5, 8], [22, 1, 0, 20, 0, 0], [-1, 0, 11, 0, 1, -5], [0, -24, 4, 4, -32, 24], [0, 8, -21, -21, -38, -8], [-5, 0, -21, 0, 5, -12], [0, -4, 2, 2, -28, 4], [47, -11, 0, 6, 0, 0], [0, 14, 19, 19, -8, -14], [0, 1, -24, -24, -21, -1], [27, 0, -23, 0, -27, 2], [29, 24, 0, 4, 0, 0], [-4, -30, 0, 4, 0, 0], [14, -19, 0, 10, 0, 0], [-8, 2, 0, -19, 0, 0], [-11, -2, 0, -13, 0, 0], [31, 0, -15, 0, -31, 5], [0, -12, 10, 10, 26, 12], [-6, 6, 0, -3, 0, 0], [-25, 0, -8, 0, 25, -7], [20, 0, 4, 0, -20, -17], [0, 15, -26, -26, -6, -15], [29, 0, 1, 0, -29, -2], [0, 12, -4, -4, -38, -12], [0, -15, 10, 10, -33, 15], [-9, 0, 17, 0, 9, 7], [0, 23, -18, -18, -1, -23], [0, 0, -2, -2, -16, 0], [38, -19, 0, -8, 0, 0], [-39, 9, 0, 9, 0, 0], [4, 1, 0, -15, 0, 0], [-57, 0, 6, 0, 57, 6], [-48, 0, -11, 0, 48, -8], [4, 0, 9, 0, -4, -26], [0, 16, 13, 13, -18, -16], [-57, 0, 1, 0, 57, 0], [0, -21, 2, 2, -28, 21], [0, 19, 20, 20, 15, -19], [20, 8, 0, -8, 0, 0], [0, 12, 0, 0, 27, -12], [4, 0, 10, 0, -4, 10], [12, 0, 9, 0, -12, -6], [0, 19, 7, 7, 3, -19], [-18, 0, -13, 0, 18, 2], [1, -29, 0, 2, 0, 0], [43, 0, 0, -3, 0, 0], [-44, 0, -14, 0, 44, -14], [0, 3, 9, 9, 3, -3], [43, 0, 6, 0, -43, 10], [-32, 0, 13, 0, 32, -21], [0, 6, -9, -9, 18, -6], [13, 0, -11, 0, -13, 26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2646_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_e_o();".
// 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_e_o(:prec:=6)
    chi := MakeCharacter_2646_e();
    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_e_o();".
// 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_e_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2646_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -12, 31, 26, 21, 5, 1]>,<11,R![1089, 858, 643, 92, 27, -1, 1]>,<13,R![9, -9, 15, 0, 7, -2, 1]>],Snew);
    return Vf;
end function;