
// Make newform 280.2.bj.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_280_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_280_bj_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_280_2_bj_d();".
// 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_280_2_bj_d();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_280_bj();"
function MakeCharacter_280_bj()
    N := 280;
    order := 6;
    char_gens := [71, 141, 57, 241];
    v := [3, 3, 6, 5];
    // 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_280_bj_Hecke();"
function MakeCharacter_280_bj_Hecke(Kf)
    N := 280;
    order := 6;
    char_gens := [71, 141, 57, 241];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [1, 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 := [[0, 1, 1, -1], [-2, 0, 1, 0], [1, 0, -1, 0], [0, -1, 0, 3], [0, 2, -2, 2], [3, 2, 0, -1], [2, -3, -1, 3], [-3, -3, -3, 0], [4, -1, 4, 0], [2, 0, -4, -7], [0, 3, -3, 3], [0, 2, 0, 0], [3, 0, -6, 0], [-3, 8, 0, -4], [6, -2, -6, 4], [6, -5, -3, 5], [2, 3, -1, -3], [6, -1, -6, 2], [0, -6, -1, -6], [-1, 0, 2, -7], [-10, -3, 5, 3], [-1, -3, -1, 0], [-5, 0, 10, 6], [5, 6, 5, 0], [-2, 0, 4, 0], [0, 3, 6, 3], [-6, 5, 6, -10], [-7, 0, 7, 0], [-12, -7, 6, 7], [-5, 2, 0, -1], [2, 0, -4, -12], [0, -12, 0, 0], [0, 6, 2, 6], [-2, 0, 4, 6], [2, -1, 2, 0], [6, -11, -3, 11], [0, -2, 6, -2], [10, 0, -10, 0], [12, -10, 0, 5], [12, -6, -12, 12], [0, -3, -13, -3], [0, 2, 0, -1], [6, 14, 6, 0], [0, -8, 0, -8], [-4, 0, 8, -2], [0, 0, -18, 0], [-3, -18, 0, 9], [0, -16, 0, 8], [8, -12, -4, 12], [-12, -2, 12, 4], [7, -1, -7, 2], [-3, 0, 6, -7], [-16, 6, 8, -6], [-3, 0, 6, -21], [0, 6, 0, 0], [-8, 23, 4, -23], [0, -5, 0, -5], [9, 3, -9, -6], [6, -3, -3, 3], [8, 16, 0, -8], [4, 0, -2, 0], [0, -16, 0, 8], [-7, 0, 14, -6], [0, 3, -9, 3], [-10, 0, -10, 0], [1, -11, 1, 0], [21, 1, -21, -2], [-3, -30, 0, 15], [0, -14, 7, -14], [12, 26, 0, -13], [-2, -33, 1, 33], [5, 19, 5, 0], [0, 1, 18, 1], [4, -6, 4, 0], [3, -34, 0, 17], [-12, -3, 12, 6], [8, -32, -4, 32], [-6, -12, 6, 24], [-5, -2, 5, 4], [10, 12, -5, -12], [-10, 0, 20, -6], [14, 0, -28, 3], [0, 8, 0, -8], [15, 0, -30, -3], [0, -8, 0, 16], [-7, 0, 7, 0], [23, -4, 0, 2], [-6, -20, 6, 40], [-24, 16, 0, -8], [-6, 0, 12, 9], [-1, 30, -1, 0], [0, -5, 3, -5], [-24, -4, 12, 4], [-25, -2, 0, 1], [-12, -6, 12, 12], [6, -10, 0, 5], [6, 13, -6, -26], [28, -12, -14, 12], [0, 0, 0, 0], [-14, -21, -14, 0], [1, 0, 0, 0], [32, 14, -16, -14], [-26, 6, 13, -6], [-10, 2, 10, -4], [0, -9, 13, -9], [-8, 24, 4, -24], [-2, 0, 4, 6], [-1, -39, -1, 0], [20, -2, -10, 2], [-6, 0, 12, -12], [-12, -11, 12, 22], [-2, 27, 1, -27], [22, 8, 0, -4], [-12, 18, 6, -18], [-2, 0, 4, 18], [0, -2, -31, -2], [6, 0, -12, 30], [0, 9, 12, 9], [-15, 7, -15, 0], [-23, 6, 0, -3], [0, -9, 12, -9], [18, 24, 0, -12], [-21, -9, 21, 18], [0, -8, -11, -8], [6, 12, 6, 0], [-12, 0, 24, 13], [18, -5, 18, 0], [36, 4, -36, -8], [6, 22, 0, -11], [0, -6, 0, 12], [0, 6, -16, 6], [-6, 0, 12, -1], [18, 16, 18, 0], [-3, 0, 6, 13], [-24, 6, -24, 0], [6, 0, -12, 0], [0, -9, -33, -9], [46, -6, -23, 6], [12, -20, 0, 10], [0, 10, 1, 10], [-5, 0, 10, 9], [0, -2, 0, 0], [-4, 27, 2, -27], [5, -16, 0, 8], [0, -12, 36, -12], [48, -4, 0, 2], [-21, 26, 0, -13], [-32, 18, 16, -18], [-12, -30, -12, 0], [22, -17, 22, 0], [9, 27, 9, 0], [5, 0, -10, -18], [0, -4, 0, 2], [42, -1, -42, 2], [0, 18, 3, 18], [-16, 0, 32, -26], [-15, -7, -15, 0], [-11, -12, -11, 0], [-8, 0, 16, 0], [0, 4, 18, 4], [-17, 18, 17, -36], [-8, 44, 0, -22], [-18, 0, 36, 3], [6, 36, 6, 0], [0, 8, 28, 8], [0, -5, -36, -5], [-42, 23, 21, -23], [0, -7, -33, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_280_bj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_280_2_bj_d();".
// 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_280_2_bj_d(:prec:=4)
    chi := MakeCharacter_280_bj();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_280_2_bj_d();".
// 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_280_2_bj_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_280_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 3, 1]>,<13,R![6, -6, 1]>],Snew);
    return Vf;
end function;