
// Make newform 170.2.h.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_170_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_170_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_170_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_170_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, 0, 0, 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_170_h();"
function MakeCharacter_170_h()
    N := 170;
    order := 4;
    char_gens := [137, 71];
    v := [4, 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_170_h_Hecke();"
function MakeCharacter_170_h_Hecke(Kf)
    N := 170;
    order := 4;
    char_gens := [137, 71];
    char_values := [[1, 0, 0, 0], [0, 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, 0, -1, 0], [1, 1, 1, 0], [0, 1, 0, 0], [1, 0, -1, 2], [1, 0, -1, 0], [1, 0, 0, 0], [-2, -3, 2, 0], [0, -1, -1, -1], [-2, 0, 2, -6], [-2, -7, -2, 0], [-3, -1, -3, 0], [3, -2, 3, 0], [-1, 0, 1, 8], [0, 3, 6, 3], [-1, 3, 0, -3], [0, -6, 5, -6], [0, -7, -1, -7], [2, 0, -2, 11], [6, 2, 0, -2], [-9, 1, -9, 0], [2, 5, 2, 0], [4, 0, -4, -6], [0, 7, 0, 7], [9, 4, 0, -4], [8, 3, 8, 0], [-10, 4, 0, -4], [-8, 4, 0, -4], [-6, -4, -6, 0], [-2, 0, 2, 9], [6, 0, -6, 5], [0, -9, 5, -9], [7, 10, 7, 0], [-4, -9, 0, 9], [5, -8, 5, 0], [-6, 3, 0, -3], [0, 2, 2, 2], [0, 2, 0, -2], [6, 0, -6, 8], [-2, 18, -2, 0], [8, -10, 8, 0], [0, 2, 4, 2], [4, 0, -4, -4], [-4, 3, 0, -3], [0, 0, 0, -4], [-11, 0, 11, 10], [-13, 7, -13, 0], [2, 0, -2, -2], [0, 13, 7, 13], [-11, 0, 11, 5], [0, -3, -12, -3], [-2, -9, -2, 0], [26, 1, 0, -1], [-17, -4, -17, 0], [12, -10, 0, 10], [0, 1, -24, 1], [0, -9, -17, -9], [-6, 5, -6, 0], [-12, 4, 0, -4], [-4, 14, -4, 0], [0, -6, 7, -6], [-1, 0, 1, -11], [-3, 6, 0, -6], [0, 7, 0, -7], [-4, 2, -4, 0], [-2, 0, 2, -12], [-16, 0, 16, 12], [0, -11, 7, -11], [-8, -19, -8, 0], [3, 0, -3, 19], [0, 12, -8, 12], [-6, 8, 0, -8], [0, -13, 6, -13], [-10, 0, 10, -22], [-4, 20, 0, -20], [-4, 0, -4, 0], [0, 15, 1, 15], [0, -16, 12, -16], [-8, 0, 8, 12], [12, 0, -12, -4], [3, 6, 0, -6], [14, 0, -14, 6], [30, -5, 0, 5], [16, 0, -16, 0], [0, 5, -8, 5], [0, -32, 0, 0], [-20, 0, 0, 0], [-11, 0, 11, 26], [0, -11, -12, -11], [0, -12, 10, -12], [-11, 7, 0, -7], [0, 15, 4, 15], [7, 17, 7, 0], [13, 0, -13, -2], [0, 9, -21, 9], [-15, 0, 15, 6], [12, 0, -12, -20], [32, 0, 0, 0], [16, 0, -16, 16], [2, 2, 0, -2], [0, -16, 0, 0], [15, -5, 15, 0], [5, 2, 0, -2], [0, -10, -18, -10], [0, 12, 9, 12], [-1, 0, 1, -34], [24, -4, 0, 4], [0, 2, -6, 2], [0, -20, 14, -20], [6, 26, 0, -26], [16, 0, -16, 4], [3, -10, 3, 0], [5, -20, 0, 20], [14, -19, 14, 0], [-6, 0, 6, 6], [0, 4, 44, 4], [-4, -10, -4, 0], [0, -26, 0, 0], [27, 15, 0, -15], [15, 0, -15, 12], [-15, 3, 0, -3], [0, -8, 0, -8], [-12, 0, 12, -5], [-29, -4, -29, 0], [-1, 13, -1, 0], [-21, 0, 21, 4], [-8, 12, 0, -12], [4, -29, 4, 0], [-19, -25, -19, 0], [-1, -15, 0, 15], [0, -2, -8, -2], [0, 19, -21, 19], [-11, 22, -11, 0], [-17, 21, -17, 0], [0, 2, 37, 2], [12, 22, 0, -22], [9, 10, 0, -10], [0, 6, -32, 6], [-11, 15, -11, 0], [0, 16, 4, 16], [23, 0, -23, -4], [29, 10, 29, 0], [18, -31, 18, 0], [10, 0, -10, 2], [-4, 0, 4, -20], [0, -8, 0, 8], [-11, 0, 11, -3], [-9, 12, -9, 0], [-2, 0, 2, 19], [0, -19, -21, -19], [24, -18, 0, 18], [-13, 0, 13, 30], [4, 34, 4, 0], [-6, -35, 0, 35], [27, 14, 27, 0], [15, 0, -15, 17], [8, 0, -8, 16], [-2, -29, 0, 29], [5, 0, -5, -12], [0, 31, 2, 31], [-8, 0, 8, 43], [-7, -11, -7, 0], [-22, 1, 0, -1], [0, -28, 6, -28], [0, -29, 19, -29], [0, -3, -30, -3], [-1, -10, -1, 0], [3, 27, 3, 0], [5, 0, -5, -40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_170_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_170_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_170_2_h_a(:prec:=4)
    chi := MakeCharacter_170_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_170_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_170_2_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_170_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -4, 8, -4, 1]>],Snew);
    return Vf;
end function;