
// Make newform 1680.2.t.i in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1680_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1680_t_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_t_i();".
// 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_t_i();".
// 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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [13, -32, -30, -5, 10, -7], [-3, -74, -32, 10, 3, -9], [-11, -72, -10, -17, 11, -10], [-65, 68, 12, 25, -27, 12], [55, -54, -42, -30, 37, -19]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_t();"
function MakeCharacter_1680_t()
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    v := [2, 2, 2, 1, 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_1680_t_Hecke();"
function MakeCharacter_1680_t_Hecke(Kf)
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, -1, 0, 0, 0, 0], [0, 0, -1, 1, 1, 1], [0, 0, 0, 1, -1, 0], [0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 0, -1], [0, 2, 0, -1, 1, 0], [-2, 0, 2, -2, -2, -2], [0, 0, 1, 0, 0, -1], [0, 0, 0, 2, -2, 0], [4, 0, 1, 0, 0, -1], [0, 4, 2, 0, 0, 2], [0, -4, 2, -2, 2, 2], [0, -4, 2, 1, -1, 2], [4, 0, -2, 2, 2, 2], [2, 0, -2, -2, -2, 2], [0, 0, 0, 0, 0, 0], [0, 0, 3, -1, -1, -3], [0, -4, 2, -3, 3, 2], [-8, 0, 2, 0, 0, -2], [0, 4, -2, 2, -2, -2], [0, 0, -1, 2, 2, 1], [0, -4, -2, 1, -1, -2], [8, 0, 3, 2, 2, -3], [0, 8, 2, 0, 0, 2], [0, -6, -4, 1, -1, -4], [2, 0, 2, -2, -2, -2], [0, -8, 2, -1, 1, 2], [0, -4, 2, 4, -4, 2], [4, 0, 0, 0, 0, 0], [0, 0, 4, -1, 1, 4], [-4, 0, -1, -2, -2, 1], [-2, 0, 4, 2, 2, -4], [16, 0, 0, -2, -2, 0], [0, 0, 2, -3, 3, 2], [0, 8, -2, 2, -2, -2], [0, 4, 0, 4, -4, 0], [0, -8, -3, -2, 2, -3], [0, 0, -3, -1, -1, 3], [2, 0, 0, -2, -2, 0], [8, 0, -3, 1, 1, 3], [0, -4, 0, 2, -2, 0], [0, 12, -2, 5, -5, -2], [-12, 0, 1, -2, -2, -1], [4, 0, 2, -6, -6, -2], [0, 8, 4, 0, 0, 4], [0, 0, -2, -4, 4, -2], [10, 0, -4, -2, -2, 4], [0, -16, 0, -1, 1, 0], [0, 0, 5, 1, 1, -5], [10, 0, -6, 4, 4, 6], [-4, 0, -2, -2, -2, 2], [0, 0, -3, -6, 6, -3], [0, -2, -4, 5, -5, -4], [-12, 0, -1, 2, 2, 1], [16, 0, 3, 0, 0, -3], [0, 0, 0, 2, -2, 0], [10, 0, -2, -4, -4, 2], [0, 4, -6, 0, 0, -6], [0, 0, -1, -2, 2, -1], [0, -4, -2, 0, 0, -2], [8, 0, 0, 6, 6, 0], [0, -12, -4, -1, 1, -4], [0, 4, -2, -5, 5, -2], [20, 0, -2, 2, 2, 2], [0, 16, 0, 0, 0, 0], [0, 10, 0, -3, 3, 0], [6, 0, -2, -6, -6, 2], [0, -8, -3, 4, -4, -3], [-8, 0, -1, -1, -1, 1], [0, 8, 0, -4, 4, 0], [0, 4, 8, -2, 2, 8], [-4, 0, 0, -2, -2, 0], [0, -8, 4, -2, 2, 4], [-6, 0, -2, 2, 2, 2], [0, 0, -6, 7, -7, -6], [6, 0, -6, 4, 4, 6], [2, 0, 6, 0, 0, -6], [20, 0, -4, 2, 2, 4], [6, 0, 2, -6, -6, -2], [32, 0, -1, -1, -1, 1], [0, 4, 0, 3, -3, 0], [-20, 0, 1, 2, 2, -1], [0, -10, 0, -5, 5, 0], [-2, 0, -4, -8, -8, 4], [0, 8, 4, -2, 2, 4], [28, 0, -9, 4, 4, 9], [0, 12, -6, 8, -8, -6], [0, 0, 2, 4, -4, 2], [-8, 0, 0, 10, 10, 0], [0, 16, -6, 6, -6, -6], [-8, 0, -5, 9, 9, 5], [-12, 0, -4, 2, 2, 4], [0, -16, 4, 2, -2, 4], [4, 0, -3, 2, 2, 3], [24, 0, 1, -4, -4, -1], [0, 20, -4, 0, 0, -4], [2, 0, 6, -2, -2, -6], [0, 8, 0, 4, -4, 0], [0, 12, -6, 3, -3, -6], [0, -16, 4, -10, 10, 4], [-22, 0, -2, 0, 0, 2], [28, 0, -4, 0, 0, 4], [0, 4, -4, -7, 7, -4], [0, 16, 0, 6, -6, 0], [0, 8, -1, -4, 4, -1], [-24, 0, 3, -1, -1, -3], [22, 0, 0, 4, 4, 0], [0, -24, -4, 0, 0, -4], [0, 28, -4, 0, 0, -4], [0, 8, 0, 5, -5, 0], [4, 0, -1, 2, 2, 1], [0, 0, -2, 0, 0, 2], [-14, 0, 2, -8, -8, -2], [0, -4, -2, 0, 0, -2], [0, 20, 4, 4, -4, 4], [0, 4, -4, -9, 9, -4], [32, 0, -5, 1, 1, 5], [2, 0, 6, 2, 2, -6], [0, -12, 0, -12, 12, 0], [0, 0, 7, -6, 6, 7], [0, -10, 4, 3, -3, 4], [4, 0, 3, -2, -2, -3], [2, 0, 2, -10, -10, -2], [6, 0, 0, 8, 8, 0], [0, 0, -2, -4, -4, 2], [0, 0, 10, -4, 4, 10], [0, 8, 4, 9, -9, 4], [-20, 0, 8, -4, -4, -8], [0, -38, 0, -1, 1, 0], [0, 0, 10, -6, -6, -10], [0, 12, 4, 2, -2, 4], [0, 0, 7, 10, 10, -7], [2, 0, 8, 4, 4, -8], [0, -24, 3, -2, 2, 3], [0, 12, -4, -4, 4, -4], [0, -8, 3, -6, 6, 3], [2, 0, 8, -12, -12, -8], [-28, 0, 1, 2, 2, -1], [14, 0, 4, 6, 6, -4], [0, -12, -6, 0, 0, -6], [0, 26, -8, 9, -9, -8], [-2, 0, -8, 10, 10, 8], [-8, 0, 2, -10, -10, -2], [0, 16, 10, -3, 3, 10], [0, 8, 1, 6, -6, 1], [24, 0, -13, 8, 8, 13], [0, 2, -4, 11, -11, -4], [0, -28, 4, -4, 4, 4], [-12, 0, 7, -2, -2, -7], [0, -12, 8, -2, 2, 8], [0, 12, -6, 2, -2, -6], [0, 12, 8, 2, -2, 8], [-32, 0, -3, 1, 1, 3], [-8, 0, 6, 10, 10, -6], [8, 0, 9, 2, 2, -9], [0, -4, 8, -9, 9, 8], [32, 0, -3, 4, 4, 3], [0, -10, 8, -1, 1, 8], [0, 0, 2, 11, -11, 2], [0, 28, -4, -2, 2, -4], [20, 0, -2, 0, 0, 2], [0, 8, -8, 9, -9, -8], [0, -24, -2, 4, -4, -2], [32, 0, -6, 2, 2, 6], [0, 0, -10, 3, -3, -10], [6, 0, -4, 8, 8, 4], [0, 16, 7, 6, -6, 7], [16, 0, 5, 7, 7, -5], [22, 0, -2, 2, 2, 2], [16, 0, -9, 7, 7, 9], [0, 12, 0, -2, 2, 0], [-12, 0, -9, 6, 6, 9], [-8, 0, 1, 8, 8, -1], [-12, 0, 14, -4, -4, -14], [-50, 0, -2, -2, -2, 2], [0, 16, 6, -8, 8, 6], [18, 0, -2, -6, -6, 2], [0, 4, 8, 2, -2, 8], [-52, 0, -2, 0, 0, 2], [0, 0, 8, 0, 0, 8], [0, -32, 7, 0, 0, 7], [0, -18, 4, -15, 15, 4], [-8, 0, 3, -2, -2, -3], [0, -28, -4, 0, 0, -4], [0, -12, 12, -12, 12, 12], [2, 0, 8, 0, 0, -8], [24, 0, -2, 4, 4, 2], [0, -12, 2, -17, 17, 2], [0, -26, 0, -13, 13, 0], [4, 0, 16, 6, 6, -16], [-12, 0, 9, -4, -4, -9], [0, 6, -4, 3, -3, -4], [0, 8, 7, -8, 8, 7], [-38, 0, -4, 0, 0, 4], [0, 32, -8, 4, -4, -8], [0, -32, 5, -10, 10, 5], [0, -12, -12, 8, -8, -12], [18, 0, -8, 2, 2, 8], [4, 0, -1, 10, 10, 1], [0, -16, 4, 9, -9, 4], [-2, 0, -10, 0, 0, 10], [28, 0, 6, -8, -8, -6], [0, 40, -1, 10, -10, -1], [-12, 0, -1, -6, -6, 1], [0, 6, -8, 3, -3, -8], [2, 0, 8, 0, 0, -8], [-40, 0, 1, -4, -4, -1], [0, -16, 0, 6, -6, 0], [-20, 0, 15, -10, -10, -15], [0, 0, -12, 0, 0, -12], [0, -20, 2, -6, 6, 2], [-8, 0, -2, -2, -2, 2], [-18, 0, 4, 4, 4, -4], [0, -16, -6, -6, 6, -6], [28, 0, -7, -6, -6, 7], [0, 10, -8, 3, -3, -8], [0, -28, 0, 3, -3, 0], [-38, 0, 8, 0, 0, -8], [-16, 0, -1, 0, 0, 1], [18, 0, -4, -4, -4, 4], [0, 0, 0, 0, 0, 0], [0, -44, 4, -12, 12, 4], [-10, 0, 8, -8, -8, -8], [0, 16, 15, -6, 6, 15], [-32, 0, -5, -1, -1, 5], [0, -16, -10, -4, 4, -10], [0, 0, -5, -3, -3, 5], [0, 44, -4, -6, 6, -4], [40, 0, -3, 4, 4, 3], [-8, 0, -8, -6, -6, 8], [-50, 0, 0, 0, 0, 0], [0, 28, -2, 12, -12, -2], [0, 8, -2, 4, -4, -2], [10, 0, -14, 12, 12, 14], [0, -4, 2, -1, 1, 2], [8, 0, -11, 11, 11, 11], [0, 0, -6, -12, -12, 6], [0, 18, 8, 1, -1, 8], [-4, 0, -3, -14, -14, 3], [0, -8, -8, -4, 4, -8], [-6, 0, -4, -4, -4, 4], [0, 8, -5, -6, 6, -5], [16, 0, -10, 2, 2, 10], [0, -24, -2, 4, -4, -2], [-4, 0, 8, -2, -2, -8], [12, 0, 2, -8, -8, -2], [0, -6, -8, 11, -11, -8], [0, 12, 4, -4, 4, 4], [-44, 0, -3, 2, 2, 3], [0, -14, -4, 7, -7, -4], [-14, 0, -6, 8, 8, 6], [0, -56, 3, -6, 6, 3], [8, 0, -7, -5, -5, 7], [-30, 0, -4, 12, 12, 4], [0, 20, -6, 20, -20, -6], [0, 24, -7, 10, -10, -7], [0, 12, 0, -15, 15, 0], [0, 12, -2, 12, -12, -2], [0, 58, 0, 1, -1, 0], [10, 0, -16, 2, 2, 16], [0, -8, -10, 11, -11, -10], [0, -24, -3, 0, 0, -3], [28, 0, 1, -2, -2, -1], [34, 0, -12, 2, 2, 12], [56, 0, -7, 4, 4, 7], [0, 32, 12, 0, 0, 12], [0, -28, -8, -5, 5, -8], [-2, 0, 2, -14, -14, -2], [0, 4, 2, 4, -4, 2], [0, 12, -10, 15, -15, -10], [8, 0, 6, -18, -18, -6], [0, 12, 10, -11, 11, 10], [0, 48, -2, 0, 0, -2], [0, 18, 0, 1, -1, 0], [-14, 0, 2, 6, 6, -2], [10, 0, -6, -6, -6, 6], [-20, 0, 6, -2, -2, -6], [0, 40, -2, 0, 0, -2], [24, 0, 2, -4, -4, -2], [0, -16, -18, 4, -4, -18], [-2, 0, -2, 2, 2, 2], [0, -28, -8, 8, -8, -8], [16, 0, -3, -7, -7, 3], [0, -16, -4, 2, -2, -4], [0, 52, 6, -1, 1, 6], [8, 0, -5, -4, -4, 5], [24, 0, 1, -10, -10, -1], [-10, 0, 0, 18, 18, 0], [0, -28, 6, 2, -2, 6], [0, -8, 0, -9, 9, 0], [-60, 0, 8, -2, -2, -8], [0, -60, 0, 2, -2, 0], [-12, 0, 17, -8, -8, -17], [-4, 0, 5, -18, -18, -5], [0, -48, 3, -12, 12, 3], [-12, 0, 2, -6, -6, -2], [0, -36, 8, -24, 24, 8], [0, -36, 0, 1, -1, 0], [0, -4, -16, 5, -5, -16], [16, 0, -12, 4, 4, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1680_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1680_2_t_i();".
// 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_t_i(:prec:=6)
    chi := MakeCharacter_1680_t();
    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_t_i();".
// 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_t_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-8, -20, 2, 1]>,<13,R![64, 0, 176, 0, 28, 0, 1]>,<19,R![8, -12, -2, 1]>],Snew);
    return Vf;
end function;