
// Make newform 1680.2.f.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1680_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1680_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_1680_2_f_e();".
// 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_1680_2_f_e();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, 1, 1, 0], [0, 2, 0, 1], [-1, 1, -1, 0]];
    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_1680_f();"
function MakeCharacter_1680_f()
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    v := [2, 2, 1, 2, 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_1680_f_Hecke();"
function MakeCharacter_1680_f_Hecke(Kf)
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [-1, 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], [0, 1, 1, 0], [-1, 0, 0, 0], [-1, 1, 2, 0], [0, 2, 2, 2], [0, -1, -2, -1], [-2, 1, 0, -1], [0, -2, 2, -2], [0, 0, 4, 0], [0, 3, 2, 3], [0, -1, -6, -1], [4, 1, 0, -1], [-4, -2, 0, 2], [-8, -4, 0, 4], [0, 2, 0, -2], [0, -2, -6, -2], [-2, -2, 0, 2], [0, 1, -4, 1], [12, 0, 0, 0], [0, -1, 4, -1], [0, -1, 8, -1], [-4, -4, 0, 4], [-8, -7, 0, 7], [12, 2, 0, -2], [0, -1, -4, -1], [6, -2, 0, 2], [0, 7, 8, 7], [0, 6, 8, 6], [-2, -2, 0, 2], [0, 4, 10, 4], [10, 3, 0, -3], [-14, -6, 0, 6], [0, 2, 10, 2], [0, -2, -18, -2], [0, -3, -14, -3], [-8, 4, 0, -4], [0, -1, 6, -1], [4, -2, 0, 2], [12, 4, 0, -4], [-10, -6, 0, 6], [0, -4, 2, -4], [0, -5, 0, -5], [0, 1, -4, 1], [-6, 0, 0, 0], [0, 2, 10, 2], [0, 5, -6, 5], [8, 0, 0, 0], [0, -3, 8, -3], [4, 1, 0, -1], [0, 1, 12, 1], [0, 2, 18, 2], [0, 5, 16, 5], [0, 8, 8, 8], [-6, 2, 0, -2], [2, 5, 0, -5], [0, -4, 0, -4], [2, 2, 0, -2], [0, -3, 22, -3], [12, 9, 0, -9], [0, 0, 20, 0], [0, -1, 10, -1], [22, 6, 0, -6], [0, 5, -2, 5], [24, 2, 0, -2], [0, -3, -4, -3], [0, 4, 2, 4], [-16, -4, 0, 4], [2, -6, 0, 6], [0, 2, 4, 2], [0, 5, -4, 5], [18, -3, 0, 3], [0, -5, -4, -5], [0, 5, 8, 5], [-16, 5, 0, -5], [-8, -8, 0, 8], [12, 8, 0, -8], [0, -5, 14, -5], [0, 9, 6, 9], [0, -2, 8, -2], [0, 6, 12, 6], [-2, -12, 0, 12], [-10, -2, 0, 2], [0, -11, 4, -11], [0, 7, -4, 7], [0, -5, -6, -5], [0, -2, 12, -2], [0, 2, -4, 2], [14, -4, 0, 4], [-6, -4, 0, 4], [6, -5, 0, 5], [-24, 3, 0, -3], [12, -8, 0, 8], [18, 11, 0, -11], [0, 6, 6, 6], [8, 8, 0, -8], [4, 10, 0, -10], [-22, -2, 0, 2], [24, -4, 0, 4], [0, 15, 6, 15], [18, -4, 0, 4], [16, -6, 0, 6], [0, -2, -14, -2], [24, -5, 0, 5], [0, 2, 16, 2], [20, 12, 0, -12], [0, -5, -8, -5], [-20, -3, 0, 3], [-26, -17, 0, 17], [0, -1, -40, -1], [0, -2, -12, -2], [0, 1, -16, 1], [-4, -13, 0, 13], [0, 0, -2, 0], [0, 0, 14, 0], [-20, -8, 0, 8], [0, -6, 4, -6], [0, -3, -22, -3], [-4, -12, 0, 12], [0, -8, -2, -8], [0, 2, 38, 2], [0, -3, -28, -3], [26, 2, 0, -2], [-18, 0, 0, 0], [0, 0, -36, 0], [0, -4, -34, -4], [0, 13, -2, 13], [-22, -2, 0, 2], [-28, 2, 0, -2], [0, 3, 36, 3], [0, 13, 22, 13], [4, 4, 0, -4], [0, 2, -4, 2], [-28, -6, 0, 6], [20, 7, 0, -7], [-20, -8, 0, 8], [0, 0, 36, 0], [-30, -6, 0, 6], [0, 3, 46, 3], [-2, 6, 0, -6], [0, -10, 4, -10], [0, -12, -2, -12], [0, 3, -22, 3], [6, -5, 0, 5], [0, -12, 0, -12], [0, -5, -24, -5], [12, 2, 0, -2], [0, 3, 2, 3], [34, 7, 0, -7], [0, -4, -30, -4], [0, 2, -24, 2], [16, -7, 0, 7], [4, 6, 0, -6], [-12, -8, 0, 8], [-24, -2, 0, 2], [24, 2, 0, -2], [0, 7, -28, 7], [-20, -20, 0, 20], [-12, 8, 0, -8], [0, 13, 0, 13], [-22, 0, 0, 0], [0, 10, -28, 10], [0, 6, 42, 6], [-22, -19, 0, 19], [-58, 0, 0, 0], [0, -6, -18, -6], [-32, -6, 0, 6], [-24, -2, 0, 2], [0, 1, -22, 1], [-2, -12, 0, 12], [26, 10, 0, -10], [0, 4, -10, 4], [0, 11, 16, 11], [0, -7, -32, -7], [-26, 0, 0, 0], [0, 1, 30, 1], [-8, 2, 0, -2], [24, 16, 0, -16], [0, -11, -6, -11], [0, -5, 16, -5], [0, -23, -32, -23], [14, 7, 0, -7], [22, 10, 0, -10], [36, -3, 0, 3], [2, -15, 0, 15], [0, -10, 4, -10], [62, 2, 0, -2], [-16, -19, 0, 19], [0, 7, -2, 7], [42, 2, 0, -2], [-12, -4, 0, 4], [0, 13, 32, 13], [0, -16, -32, -16], [28, 20, 0, -20], [-42, -20, 0, 20], [0, -16, -4, -16], [22, 11, 0, -11], [22, 20, 0, -20], [0, 11, 0, -11], [-14, -11, 0, 11], [-36, -10, 0, 10], [0, 15, 34, 15], [0, -7, -42, -7], [0, 19, 46, 19], [0, -10, 16, -10], [18, -2, 0, 2], [26, -6, 0, 6], [0, -11, -42, -11], [0, -12, -8, -12], [0, 12, -24, 12], [0, 2, 42, 2], [14, -4, 0, 4], [38, 10, 0, -10], [14, 23, 0, -23], [44, 11, 0, -11], [16, 6, 0, -6], [0, -8, -8, -8], [-2, -9, 0, 9], [-12, -20, 0, 20], [0, 12, 24, 12], [0, -22, -26, -22], [-14, 14, 0, -14], [0, -11, -2, -11], [0, 2, -24, 2], [42, 1, 0, -1], [20, -3, 0, 3], [-22, -2, 0, 2], [26, 5, 0, -5], [0, -19, -28, -19], [0, 7, 20, 7], [0, 6, 46, 6], [16, -3, 0, 3], [0, -10, -6, -10], [24, 6, 0, -6], [0, 0, -20, 0], [0, 13, -6, 13], [48, 0, 0, 0], [0, 2, -12, 2], [0, -30, -34, -30], [0, 4, -30, 4], [-24, -26, 0, 26], [0, 4, 48, 4], [0, 22, 22, 22], [0, 5, -4, 5], [22, 12, 0, -12], [-22, -3, 0, 3], [20, 20, 0, -20], [0, -1, 32, -1], [-14, 4, 0, -4], [8, 8, 0, -8], [0, 10, -16, 10], [-44, -17, 0, 17], [16, 8, 0, -8], [0, 6, 8, 6], [0, 20, 36, 20], [38, 12, 0, -12], [0, -10, 26, -10], [2, 10, 0, -10], [0, 7, -30, 7], [-26, 12, 0, -12], [0, 1, 28, 1], [58, 7, 0, -7], [0, 22, 44, 22], [0, -5, -44, -5], [0, -17, -38, -17], [-10, -7, 0, 7], [0, -10, 14, -10], [0, 7, -14, 7], [28, 20, 0, -20], [-60, 4, 0, -4], [0, 16, 2, 16], [0, -13, -8, -13], [-16, -8, 0, 8], [0, 19, 8, 19], [36, 6, 0, -6], [0, 13, 40, 13], [0, -5, 56, -5], [0, -12, -40, -12], [-2, -2, 0, 2], [42, 10, 0, -10], [42, 20, 0, -20], [-24, 12, 0, -12], [-24, -2, 0, 2], [8, -10, 0, 10], [0, 23, 8, 23], [0, 23, 26, 23], [0, -3, -8, -3], [-10, 6, 0, -6], [0, 20, -2, 20], [0, -9, -10, -9], [-24, -24, 0, 24], [0, 19, 34, 19], [24, 1, 0, -1], [0, 24, 30, 24], [50, 8, 0, -8], [44, -5, 0, 5], [-14, 16, 0, -16], [0, -1, 14, -1], [30, 4, 0, -4], [-34, 16, 0, -16], [0, 3, 26, 3], [0, 27, 36, 27], [0, 0, -62, 0], [-8, 2, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1680_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1680_2_f_e();".
// 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_1680_2_f_e(:prec:=4)
    chi := MakeCharacter_1680_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_1680_2_f_e();".
// 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_1680_2_f_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![20, 0, 1]>,<17,R![4, 6, 1]>,<41,R![-16, 4, 1]>],Snew);
    return Vf;
end function;