
// Make newform 287.2.h.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_287_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_287_h_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_287_2_h_a();".
// 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_287_2_h_a();".
// 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, -1, 1, -1, 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_287_h();"
function MakeCharacter_287_h()
    N := 287;
    order := 5;
    char_gens := [206, 211];
    v := [5, 3];
    // 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_287_h_Hecke();"
function MakeCharacter_287_h_Hecke(Kf)
    N := 287;
    order := 5;
    char_gens := [206, 211];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1]];
    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, 0, -2, 2], [0, 0, 0, -1], [0, 1, 0, 0], [0, -1, -2, -1], [-5, 3, -3, 5], [0, -5, 2, -5], [-4, -1, -4, 0], [2, -5, 5, -2], [-3, 3, 0, 2], [0, -3, -1, -3], [4, -4, 0, 5], [-6, 7, -2, 4], [-5, 7, -7, 5], [4, -3, 3, -4], [-4, 4, 0, 0], [-10, 7, -7, 10], [-3, 1, -3, 0], [0, 0, 0, 8], [0, -10, 2, -10], [9, 0, 7, -7], [-3, 0, 2, -2], [7, 0, 0, 0], [-11, 4, -11, 0], [-4, 4, 0, 11], [6, -1, 6, 0], [-6, -3, -6, 0], [5, -4, 5, 0], [-2, 0, -3, 3], [0, 14, -2, 14], [12, -12, 0, -3], [0, 12, -8, 12], [2, 0, -6, 6], [-11, 8, -11, 0], [8, -8, 0, 7], [3, -12, 12, -3], [-5, -4, -5, 0], [-8, 0, -5, 5], [12, 0, 8, -8], [-5, 0, -10, 10], [2, -2, 0, -18], [0, -3, 7, -3], [-9, 0, 1, -1], [8, -8, 0, -4], [0, -12, 0, -12], [-19, 23, -23, 19], [7, 9, -9, -7], [-26, 26, -26, 26], [2, 0, 2, 0], [0, -1, 2, -1], [11, -11, 11, -11], [-6, 7, -6, 0], [7, -7, 0, -5], [13, -13, 0, -4], [0, -14, 20, -14], [0, 9, -18, 9], [25, -19, 19, -25], [5, -1, 5, 0], [0, -1, 16, -1], [11, 2, -2, -11], [1, -1, 0, 9], [-6, -6, 6, 6], [-10, 28, -10, 0], [0, -13, 18, -13], [10, -10, 0, 18], [0, -2, -27, -2], [-9, 0, 0, 0], [-9, 0, 2, -2], [-21, 18, -21, 0], [-18, 11, -18, 0], [11, -29, 11, 0], [0, -4, -4, -4], [16, -13, 13, -16], [-9, 9, 0, -9], [0, 2, 9, 2], [-4, 0, 12, -12], [-1, -1, -1, 0], [1, -22, 22, -1], [-17, 0, -7, 7], [3, 0, 11, -11], [-10, 0, -18, 18], [0, -10, -16, -10], [-12, -6, -12, 0], [11, -21, 21, -11], [-8, 8, 0, -17], [0, -10, -5, -10], [-6, 17, -17, 6], [-9, 14, -14, 9], [0, 20, 4, 20], [-12, 12, 0, -6], [21, -4, 21, 0], [-12, 18, -18, 12], [-8, 8, 0, 29], [25, 0, 20, -20], [-4, 0, -4, 0], [0, -9, -8, -9], [0, 5, -10, 5], [-26, 26, 0, 3], [0, -10, 33, -10], [0, -6, -8, -6], [-19, 0, 2, -2], [0, -13, -23, -13], [0, 10, -17, 10], [-16, 16, 0, -9], [-14, 0, 12, -12], [-22, 0, 13, -13], [-14, 5, -5, 14], [-10, 6, -10, 0], [-5, 23, -5, 0], [14, 0, 20, -20], [0, -2, 31, -2], [29, -32, 32, -29], [11, -28, 28, -11], [-15, 15, 0, -14], [-25, 7, -25, 0], [-8, 8, 0, 40], [16, 4, -4, -16], [31, 0, 4, -4], [-2, 0, 17, -17], [1, 0, 18, -18], [-22, 22, 0, 30], [0, 26, -15, 26], [6, -13, 6, 0], [-3, 0, -24, 24], [3, -17, 17, -3], [18, -18, 0, -17], [7, -7, 0, 31], [34, -9, 34, 0], [0, 26, -13, 26], [-26, 26, 0, 21], [-34, 0, -28, 28], [4, -4, 0, 28], [-2, 5, -5, 2], [-9, 5, -9, 0], [20, 10, -10, -20], [0, 12, -19, 12], [-30, 35, -35, 30], [0, -30, 3, -30], [11, -15, 15, -11], [0, 10, -36, 10], [7, 0, -25, 25], [-16, 0, -25, 25], [21, 0, -11, 11], [-4, 28, -4, 0], [3, 0, 8, -8], [15, -29, 15, 0], [0, -44, 26, -44], [23, -23, 0, -1], [-1, -23, 23, 1], [-19, 14, -14, 19], [19, -27, 19, 0], [12, -22, 12, 0], [-6, 44, -6, 0], [-20, 20, 0, -5], [22, -22, 0, -12], [-28, 0, -46, 46], [0, 14, 3, 14], [-10, 0, -13, 13], [15, 13, -13, -15], [17, -39, 39, -17], [3, -3, 0, -27], [0, -20, -6, -20], [0, 17, 16, 17], [-33, 28, -28, 33], [-6, 37, -6, 0], [8, 0, -23, 23], [5, -16, 5, 0], [12, -19, 19, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_287_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_287_2_h_a();".
// 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_287_2_h_a(:prec:=4)
    chi := MakeCharacter_287_h();
    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_287_2_h_a();".
// 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_287_2_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_287_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;