
// Make newform 1690.2.e.l in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1690_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1690_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_1690_2_e_l();".
// 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_1690_2_e_l();".
// 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, 0, 1, 0], [0, 1, 0, 1], [0, 2, 0, -1]];
    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_1690_e();"
function MakeCharacter_1690_e()
    N := 1690;
    order := 3;
    char_gens := [677, 171];
    v := [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_1690_e_Hecke();"
function MakeCharacter_1690_e_Hecke(Kf)
    N := 1690;
    order := 3;
    char_gens := [677, 171];
    char_values := [[1, 0, 0, 0], [-1, 1, 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, -1, 0, 0], [0, 1, -1, 0], [-1, 0, 0, 0], [3, -3, 0, 0], [0, -3, 0, 0], [0, 0, 0, 0], [3, -3, -3, 3], [-3, 3, -2, 2], [0, 6, 2, 0], [0, 6, -2, 0], [3, 0, 0, 1], [0, 6, -3, 0], [0, 0, -6, 0], [2, -2, 0, 0], [-3, 0, 0, 0], [-3, 0, 0, 2], [0, 0, -6, 6], [-1, 1, 3, -3], [0, 0, 0, 0], [6, -6, 0, 0], [-3, 0, 0, -5], [1, 0, 0, -3], [-3, 0, 0, -3], [0, -12, 3, 0], [3, -3, 7, -7], [0, 9, 1, 0], [4, 0, 0, 3], [0, -9, -5, 0], [-12, 0, 0, 2], [0, 0, 4, -4], [0, 16, -3, 0], [-6, 0, 0, 7], [-3, 3, -3, 3], [4, -4, 3, -3], [-6, 6, 0, 0], [15, 0, 0, 5], [-13, 0, 0, 0], [-9, 9, 1, -1], [0, -3, 0, 0], [-15, 15, 0, 0], [0, -6, -2, 0], [1, 0, 0, -9], [21, -21, -1, 1], [0, 6, 6, 0], [0, 6, 9, 0], [4, -4, 6, -6], [0, 2, 9, 0], [0, -3, -2, 0], [-6, 6, 6, -6], [3, 0, 0, -1], [-3, 0, 0, -1], [-3, 0, 0, 3], [0, 0, 5, -5], [12, -12, -3, 3], [0, 0, -8, 0], [0, 12, -9, 0], [15, -15, -7, 7], [0, -6, -2, 0], [-1, 1, 0, 0], [0, 0, 0, 6], [0, -20, 6, 0], [6, -6, -3, 3], [24, 0, 0, 2], [-3, 0, 0, 7], [16, 0, 0, 6], [-18, 0, 0, -3], [-6, 6, 4, -4], [-1, 0, 0, -3], [27, -27, -1, 1], [0, -15, 11, 0], [0, 15, -3, 0], [-12, 0, 0, 6], [0, 16, -6, 0], [-10, 10, -6, 6], [0, 9, 12, 0], [-12, 12, 6, -6], [-21, 0, 0, -1], [-6, 6, 3, -3], [0, 0, 3, 0], [18, -18, 1, -1], [0, 0, 10, 0], [15, 0, 0, -7], [0, -3, -3, 0], [-2, 2, -6, 6], [0, -19, 9, 0], [-12, 0, 0, -6], [0, 0, -9, 9], [0, 27, -5, 0], [15, -15, -9, 9], [-6, 0, 0, -4], [24, 0, 0, 8], [0, 15, 15, 0], [-9, 9, -14, 14], [0, 12, 5, 0], [12, 0, 0, 8], [18, -18, -5, 5], [0, -12, -6, 0], [3, 0, 0, 2], [0, -8, 6, 0], [-3, 0, 0, -11], [14, 0, 0, -12], [0, 12, -3, 0], [18, -18, 2, -2], [0, -3, -14, 0], [2, 0, 0, 21], [3, 0, 0, -17], [0, 30, 0, 0], [0, 0, 0, 12], [-6, 0, 0, -8], [0, 17, 12, 0], [22, -22, -9, 9], [0, 6, 19, 0], [-18, 18, 0, 0], [21, 0, 0, 6], [0, 0, 12, -12], [39, -39, 4, -4], [9, -9, -1, 1], [0, -6, 3, 0], [0, 9, -6, 0], [-12, 12, -10, 10], [0, -3, -19, 0], [0, 14, 6, 0], [-12, 0, 0, -8], [0, 0, -18, 18], [0, -9, -16, 0], [-3, 0, 0, -19], [-15, 15, -15, 15], [12, -12, -8, 8], [2, 0, 0, -21], [-18, 0, 0, 19], [0, 9, -24, 0], [0, -24, 6, 0], [0, -10, -6, 0], [0, 17, 12, 0], [24, -24, 3, -3], [0, 0, -22, 0], [-18, 18, -9, 9], [-3, 3, 23, -23], [-6, 6, 4, -4], [0, -24, 0, 0], [33, 0, 0, 6], [0, 15, 3, 0], [-22, 22, 3, -3], [27, 0, 0, 9], [0, -20, 0, 0], [27, -27, 3, -3], [0, 0, 0, -8], [15, 0, 0, 1], [16, 0, 0, -21], [12, 0, 0, 18], [24, -24, -2, 2], [0, 9, 10, 0], [11, 0, 0, -9], [0, -6, 27, 0], [0, -11, 9, 0], [-18, 0, 0, -2], [-10, 10, 12, -12], [24, -24, 12, -12], [-20, 0, 0, 6], [24, 0, 0, -12], [0, 21, -21, 0], [9, -9, 9, -9], [-3, 0, 0, -24], [0, 0, 27, -27], [0, 6, 6, 0], [-3, 0, 0, -24], [0, -20, 18, 0], [1, -1, 24, -24], [-12, 0, 0, -9], [6, 0, 0, -32], [24, 0, 0, 6], [-6, 6, -20, 20], [-27, 27, 11, -11], [-12, 12, 2, -2], [22, 0, 0, 0], [51, -51, 2, -2], [0, 12, 10, 0], [-21, 0, 0, -15], [0, -34, -6, 0], [0, -10, 0, 0], [0, 0, 0, 2], [24, 0, 0, -15], [7, 0, 0, 24], [-30, 0, 0, 6], [0, 9, -6, 0], [-3, 3, -11, 11], [-11, 0, 0, 24], [-36, 0, 0, 16], [0, -12, 7, 0], [-15, 15, 9, -9], [7, -7, -9, 9], [24, -24, 0, 0], [-16, 0, 0, -9], [0, 33, 9, 0], [-36, 36, -18, 18], [0, -9, 21, 0], [-24, 0, 0, -20], [1, -1, -24, 24], [-6, 0, 0, 18], [0, 0, 0, 6], [-24, 24, -6, 6], [31, -31, -15, 15], [0, 6, 8, 0], [17, 0, 0, 18], [0, -27, 0, 0], [0, -24, -8, 0], [12, 0, 0, -4], [6, -6, 14, -14], [0, -18, 21, 0], [34, -34, 9, -9], [0, 19, -9, 0], [21, 0, 0, -9], [0, 8, 9, 0], [18, -18, 0, 0], [30, -30, 0, 0], [-6, 0, 0, 9], [22, 0, 0, 18], [9, -9, 0, 0], [0, -27, 12, 0], [36, 0, 0, 21], [0, 1, -15, 0], [24, 0, 0, -10], [0, 0, 0, -9], [36, -36, -2, 2], [0, -3, 13, 0], [16, 0, 0, 0], [0, -27, -13, 0], [-3, 3, -21, 21], [4, -4, 0, 0], [-21, 0, 0, 30], [0, 5, -30, 0], [0, -2, -33, 0], [0, -21, 19, 0], [-15, 0, 0, 24], [14, 0, 0, -6], [21, 0, 0, 12], [-6, 6, -4, 4], [0, 18, 24, 0], [66, -66, -3, 3], [0, -33, -5, 0], [0, 48, -6, 0], [0, -56, 0, 0], [-4, 4, -39, 39], [0, 18, -4, 0], [9, -9, -21, 21], [45, 0, 0, -5], [-30, 30, 12, -12], [0, 27, -18, 0], [-39, 39, 0, 0], [0, 6, 23, 0], [0, -18, 4, 0], [0, -18, -21, 0], [27, 0, 0, 18], [0, 16, 0, 0], [27, 0, 0, 2], [33, -33, 24, -24], [-7, 7, -27, 27], [0, -36, 6, 0], [3, 0, 0, 22], [27, -27, 9, -9], [44, 0, 0, -9], [0, -39, -1, 0], [30, 0, 0, 2], [0, -14, 24, 0], [12, -12, 18, -18], [32, -32, 18, -18], [-42, 42, 12, -12], [-18, 0, 0, 9], [-45, 45, 21, -21], [-42, 42, -14, 14], [-37, 0, 0, -3], [33, 0, 0, 1], [0, -45, -15, 0], [5, -5, 33, -33], [-40, 40, 12, -12], [0, -12, 2, 0], [-48, 48, -18, 18], [33, 0, 0, 13], [60, -60, -6, 6], [24, -24, 8, -8], [0, 54, -13, 0], [0, 0, 36, 0], [-6, 0, 0, 30], [0, -36, -8, 0], [45, 0, 0, -17], [9, 0, 0, -7], [-2, 0, 0, 24], [-6, 0, 0, -21], [-42, 42, 16, -16], [9, -9, 38, -38], [0, 3, -33, 0], [-15, 15, 33, -33], [0, 6, -6, 0], [45, -45, -12, 12], [28, -28, -24, 24], [-6, 0, 0, 36], [5, 0, 0, -15], [0, 9, -12, 0], [0, -24, -6, 0], [0, -15, 29, 0], [-58, 58, 6, -6], [0, 0, 0, -24], [0, 38, -6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1690_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1690_2_e_l();".
// 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_1690_2_e_l(:prec:=4)
    chi := MakeCharacter_1690_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_1690_2_e_l();".
// 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_1690_2_e_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1690_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 4, 6, -2, 1]>,<7,R![9, -3, 1]>,<11,R![9, 3, 1]>,<19,R![9, -18, 39, 6, 1]>,<31,R![6, -6, 1]>],Snew);
    return Vf;
end function;