
// Make newform 1680.2.f.j in Magma, downloaded from the LMFDB on 19 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_j();".
// 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_j();".
// 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, 4, 0, 2], [-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], [1, 1, 0, 0], [1, 0, 0, 0], [1, 0, 1, 1], [0, 1, -1, 1], [0, -1, 0, -1], [-2, -2, 0, 2], [0, 1, 3, 1], [0, 3, 2, 3], [0, 0, -2, 0], [0, -1, -1, -1], [2, 2, 0, -2], [6, -2, 0, 2], [6, 1, 0, -1], [-2, -1, 0, 1], [0, -1, -4, -1], [4, 4, 0, -4], [0, 4, 0, 4], [-6, -3, 0, 3], [0, -5, -3, -5], [0, -1, 4, -1], [-8, -2, 0, 2], [-6, -1, 0, 1], [10, 2, 0, -2], [0, 5, 4, 5], [2, 4, 0, -4], [0, 1, -6, 1], [0, 3, -2, 3], [2, -4, 0, 4], [0, -1, -8, -1], [10, 9, 0, -9], [8, 6, 0, -6], [0, 1, 0, 1], [0, 1, 7, 1], [0, -6, -4, -6], [-4, -4, 0, 4], [0, 5, -4, 5], [-10, -7, 0, 7], [-10, 1, 0, -1], [-2, 0, 0, 0], [0, 1, -7, 1], [0, -2, -2, -2], [0, -1, -3, -1], [-6, 0, 0, 0], [0, 1, 12, 1], [0, -1, 5, -1], [8, 0, 0, 0], [0, -3, -4, -3], [22, 1, 0, -1], [0, -8, -2, -8], [0, -5, 4, -5], [0, -5, 3, -5], [0, -10, -2, -10], [-8, -10, 0, 10], [14, 2, 0, -2], [0, -5, -8, -5], [-22, 2, 0, -2], [0, -9, -5, -9], [-6, 6, 0, -6], [0, -6, -2, -6], [0, -7, 0, -7], [14, 0, 0, 0], [0, 5, 6, 5], [-20, -10, 0, 10], [0, 3, -4, 3], [0, 5, 0, 5], [16, 4, 0, -4], [2, 0, 0, 0], [0, 7, 12, 7], [0, -4, -2, -4], [-18, -6, 0, 6], [0, -13, -9, -13], [0, -7, -8, -7], [10, -2, 0, 2], [-4, 2, 0, -2], [-14, -1, 0, 1], [0, 2, 0, 2], [0, -3, 0, -3], [0, -4, -12, -4], [0, 0, -6, 0], [-28, 0, 0, 0], [30, 2, 0, -2], [0, -1, 1, -1], [0, 7, 12, 7], [0, -5, 1, -5], [0, -13, -4, -13], [0, -8, -10, -8], [-14, 4, 0, -4], [22, 2, 0, -2], [22, -1, 0, 1], [-18, -15, 0, 15], [8, 4, 0, -4], [2, 1, 0, -1], [0, 9, 15, 9], [4, 10, 0, -10], [10, 7, 0, -7], [18, 4, 0, -4], [-2, 8, 0, -8], [0, -3, 6, -3], [-34, -8, 0, 8], [-14, -3, 0, 3], [0, 5, 4, 5], [2, 13, 0, -13], [0, -8, -6, -8], [8, -12, 0, 12], [0, 13, 8, 13], [-10, 3, 0, -3], [22, -2, 0, 2], [0, 7, 11, 7], [0, -14, -14, -14], [0, 13, 12, 13], [10, 10, 0, -10], [0, -15, -4, -15], [0, -3, -1, -3], [-4, -4, 0, 4], [0, 6, -10, 6], [0, -3, 14, -3], [-2, 9, 0, -9], [0, 11, 4, 11], [0, 1, -19, 1], [0, -12, -4, -12], [10, 4, 0, -4], [6, 0, 0, 0], [0, 3, -12, 3], [0, 5, 1, 5], [0, -4, 4, -4], [-18, -4, 0, 4], [-16, 8, 0, -8], [0, -3, 0, -3], [0, 7, 20, 7], [-28, -10, 0, 10], [0, 7, 8, 7], [44, 0, 0, 0], [18, 14, 0, -14], [-14, -14, 0, 14], [0, 6, 12, 6], [-18, 0, 0, 0], [0, -3, 10, -3], [14, 12, 0, -12], [0, 10, 0, 10], [0, -3, 1, -3], [0, 12, 16, 12], [-2, 11, 0, -11], [0, 3, 18, 3], [0, 10, 4, 10], [-8, -4, 0, 4], [0, -9, -4, -9], [-26, -2, 0, 2], [0, -7, 11, -7], [0, 7, 22, 7], [-30, -2, 0, 2], [2, 12, 0, -12], [22, 11, 0, -11], [-10, -5, 0, 5], [26, 1, 0, -1], [0, -13, -15, -13], [32, 2, 0, -2], [10, 8, 0, -8], [0, 1, -8, 1], [22, 18, 0, -18], [0, -1, -12, -1], [0, 3, 0, 3], [-38, -5, 0, 5], [-8, 6, 0, -6], [0, 21, 12, 21], [-22, -15, 0, 15], [-20, -16, 0, 16], [0, 13, 0, 13], [-2, -6, 0, 6], [-42, -8, 0, 8], [0, -7, 9, -7], [0, 14, 6, 14], [0, -5, -15, -5], [10, 0, 0, 0], [0, -5, 1, -5], [-18, -4, 0, 4], [-32, 8, 0, -8], [0, 2, 26, 2], [0, 13, -4, 13], [0, -2, 8, -2], [-6, -13, 0, 13], [-8, -14, 0, 14], [-30, -6, 0, 6], [22, -6, 0, 6], [0, 19, 12, 19], [-22, -4, 0, 4], [-14, -2, 0, 2], [0, 1, 2, 1], [14, 16, 0, -16], [-44, 2, 0, -2], [0, -5, -24, -5], [0, 7, -6, 7], [24, -8, 0, 8], [-34, -8, 0, 8], [0, -5, -4, -5], [-18, 14, 0, -14], [-18, -8, 0, 8], [26, -2, 0, 2], [-2, -14, 0, 14], [-38, -13, 0, 13], [0, -12, 14, -12], [0, -1, 23, -1], [0, -11, -16, -11], [0, 2, 24, 2], [20, 4, 0, -4], [-26, 12, 0, -12], [0, -5, -11, -5], [0, 3, 2, 3], [0, 6, 24, 6], [0, 5, -19, 5], [34, 16, 0, -16], [-54, -2, 0, 2], [46, 13, 0, -13], [38, -7, 0, 7], [8, 12, 0, -12], [0, -2, -34, -2], [-38, -3, 0, 3], [-22, -14, 0, 14], [0, 3, 0, 3], [0, 7, 12, 7], [-18, 10, 0, -10], [0, 19, 5, 19], [0, -2, 16, -2], [-50, 5, 0, -5], [-26, -9, 0, 9], [-6, -4, 0, 4], [-10, -22, 0, 22], [0, -5, -1, -5], [0, -11, -6, -11], [0, 9, 11, 9], [34, 18, 0, -18], [0, -19, -13, -19], [-12, -12, 0, 12], [0, 6, 14, 6], [0, -23, -14, -23], [-42, 3, 0, -3], [0, 8, 14, 8], [0, -9, 16, -9], [0, -7, 17, -7], [44, 10, 0, -10], [0, -13, -10, -13], [0, 1, -19, 1], [0, 5, -14, 5], [10, -6, 0, 6], [-26, 6, 0, -6], [56, 8, 0, -8], [0, -7, 4, -7], [-8, -2, 0, 2], [52, 10, 0, -10], [0, -1, 18, -1], [-22, -22, 0, 22], [42, 2, 0, -2], [0, -9, -2, -9], [0, 32, 14, 32], [-2, -12, 0, 12], [0, 13, 29, 13], [18, 26, 0, -26], [0, -11, -20, -11], [-34, 0, 0, 0], [0, -11, 8, -11], [14, -1, 0, 1], [0, -13, 12, -13], [0, -8, 14, -8], [0, 7, -4, 7], [14, 2, 0, -2], [0, -7, -29, -7], [0, 20, 10, 20], [30, 8, 0, -8], [-2, -1, 0, 1], [0, 11, 12, 11], [0, -10, -6, -10], [18, 11, 0, -11], [0, 7, -24, 7], [-24, 6, 0, -6], [0, -11, -28, -11], [0, 13, 12, 13], [0, 3, -14, 3], [38, 20, 0, -20], [-14, 14, 0, -14], [16, -10, 0, 10], [-30, -3, 0, 3], [28, 8, 0, -8], [14, -13, 0, 13], [0, -10, -14, -10], [0, 5, 10, 5], [0, 3, -7, 3], [2, -12, 0, 12], [0, 1, 12, 1], [0, 15, 5, 15], [-54, -24, 0, 24], [0, -16, -8, -16], [-58, -11, 0, 11], [0, 3, 12, 3], [-16, 14, 0, -14], [-6, 14, 0, -14], [-2, 16, 0, -16], [0, 23, -5, 23], [38, -8, 0, 8], [36, 28, 0, -28], [0, -3, 14, -3], [0, 15, 0, 15], [0, -3, -16, -3], [0, 10, 0, -10]];
    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_j();".
// 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_j(: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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![16, 0, 28, 0, 1]>,<17,R![-20, 0, 1]>,<41,R![44, -16, 1]>],Snew);
    return Vf;
end function;