
// Make newform 85.2.b.a in Magma, downloaded from the LMFDB on 13 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_85_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_85_b_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_85_2_b_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_85_2_b_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, 2, 2, -8, 14, -2, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [63, 378, -276, 944, -127, 4, 16, 64], [-67, -402, 1448, -1009, 130, 16, 64, -63], [268, -1263, 1545, -1068, 118, -64, 63, -67], [306, 1198, -1614, 1304, -70, 65, -59, 83], [-269, 619, 662, -2218, 441, 69, -43, -172], [-173, -1038, 1244, -2572, 369, -92, -49, -196], [2, 1, -8, 14, -2, 0, 0, 1]];
    Rf_basisdens := [1, 319, 319, 319, 319, 319, 319, 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_85_b();"
function MakeCharacter_85_b()
    N := 85;
    order := 2;
    char_gens := [52, 71];
    v := [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_85_b_Hecke();"
function MakeCharacter_85_b_Hecke(Kf)
    N := 85;
    order := 2;
    char_gens := [52, 71];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 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]];
    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, -1, 0], [0, 0, 1, 1, -1, 1, -1, 0], [0, -1, 0, 1, 0, 1, 0, 0], [0, 0, -3, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 1, -2, 0, -1], [0, 1, 2, -3, 3, -3, 2, 0], [0, 0, 1, 0, 0, 0, 0, 0], [-2, 0, 0, 2, 2, -2, 0, 0], [0, -2, 1, 1, -1, 1, -1, 0], [2, 0, 0, -2, 0, 0, 0, -2], [-3, 0, 0, 1, -2, 3, 0, 3], [0, 2, -2, -2, 2, -2, 0, 0], [0, 0, 0, -1, -1, 3, 0, 0], [0, 1, 4, 1, -1, 1, 0, 0], [0, -1, -2, 3, -3, 3, -2, 0], [0, 4, 2, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 3, -3, 0, -4], [0, -3, -2, 3, -3, 3, -2, 0], [-3, 0, 0, 2, -1, 2, 0, 3], [0, -4, -2, 3, -3, 3, 2, 0], [5, 0, 0, 1, 0, 5, 0, 1], [0, -1, -2, -3, 3, -3, 0, 0], [-6, 0, 0, 4, 3, 1, 0, 1], [0, 2, 0, -4, 4, -4, 6, 0], [2, 0, 0, 2, -3, 3, 0, 5], [0, -1, 8, -1, 1, -1, 2, 0], [0, 0, 5, -1, 1, -1, -5, 0], [0, 0, 0, -1, -1, -1, 0, 0], [0, 0, 4, 5, -5, 5, -8, 0], [0, -1, -8, -1, 1, -1, -4, 0], [-7, 0, 0, -5, -2, -5, 0, -3], [0, 3, 2, 3, -3, 3, -6, 0], [3, 0, 0, 8, 7, 4, 0, 1], [10, 0, 0, 4, -2, 8, 0, 6], [-2, 0, 0, -2, 0, -4, 0, -2], [0, 4, -2, -4, 4, -4, 8, 0], [0, 0, 3, -4, 4, -4, 11, 0], [0, 0, -7, 0, 0, 0, 5, 0], [0, -2, -8, 0, 0, 0, 2, 0], [-6, 0, 0, -2, -2, 0, 0, 0], [-4, 0, 0, 8, 4, -2, 0, 4], [4, 0, 0, 0, 0, 6, 0, 0], [0, -2, 4, 1, -1, 1, -4, 0], [0, -4, 2, -4, 4, -4, 0, 0], [3, 0, 0, -5, -8, -1, 0, 3], [9, 0, 0, -1, 4, 1, 0, -5], [0, -5, 6, 7, -7, 7, -8, 0], [0, 0, -5, 6, -6, 6, -1, 0], [10, 0, 0, -6, -5, 5, 0, -1], [0, 4, 2, -4, 4, -4, 0, 0], [-4, 0, 0, 8, 10, -6, 0, -2], [-6, 0, 0, 5, 5, -1, 0, 0], [8, 0, 0, 6, 6, 4, 0, 0], [0, 3, 2, -5, 5, -5, 2, 0], [0, -7, 12, 1, -1, 1, -4, 0], [0, 0, 0, 3, -9, 3, 0, 12], [8, 0, 0, 2, 2, -6, 0, 0], [0, -4, -4, 3, -3, 3, -4, 0], [-2, 0, 0, -10, -8, -2, 0, -2], [0, -2, -3, -2, 2, -2, 9, 0], [0, 2, -16, 0, 0, 0, -4, 0], [0, -1, 8, 9, -9, 9, -14, 0], [-3, 0, 0, 6, 7, 2, 0, -1], [0, -4, -2, -5, 5, -5, 6, 0], [0, 2, 22, -5, 5, -5, 2, 0], [-10, 0, 0, 2, 2, -14, 0, 0], [0, -10, -6, 4, -4, 4, 4, 0], [0, 10, 5, -12, 12, -12, 5, 0], [-2, 0, 0, 2, 10, -4, 0, -8], [0, -2, -16, 6, -6, 6, -4, 0], [-14, 0, 0, -6, -6, 4, 0, 0], [0, -8, -7, 0, 0, 0, 1, 0], [0, 11, -2, -7, 7, -7, 2, 0], [23, 0, 0, 1, -4, 7, 0, 5], [0, 9, 6, -13, 13, -13, 12, 0], [2, 0, 0, 0, 1, -7, 0, -1], [0, -2, 8, 7, -7, 7, -4, 0], [-4, 0, 0, -6, -14, 12, 0, 8], [2, 0, 0, 4, 0, 0, 0, 4], [-3, 0, 0, -19, -14, -9, 0, -5], [2, 0, 0, -6, -3, 1, 0, -3], [13, 0, 0, -7, 0, 5, 0, -7], [0, 1, -2, 11, -11, 11, -14, 0], [7, 0, 0, -4, -3, 2, 0, -1], [0, -1, -12, -7, 7, -7, -2, 0], [8, 0, 0, 5, 11, -13, 0, -6], [0, -3, 10, -5, 5, -5, -2, 0], [14, 0, 0, -6, 2, 0, 0, -8], [0, -1, 16, 7, -7, 7, -2, 0], [0, -3, 8, -5, 5, -5, 12, 0], [5, 0, 0, -3, 6, -9, 0, -9], [0, 0, -23, 0, 0, 0, -7, 0], [-10, 0, 0, 4, 4, 10, 0, 0], [-3, 0, 0, -11, -2, -1, 0, -9], [0, 2, 7, 0, 0, 0, 3, 0], [-14, 0, 0, 10, 8, -8, 0, 2], [-28, 0, 0, 0, 4, -10, 0, -4], [0, 7, 2, 9, -9, 9, -6, 0], [-18, 0, 0, -4, 0, -16, 0, -4], [0, 10, -17, -1, 1, -1, 9, 0], [0, -7, 6, 1, -1, 1, 6, 0], [0, 3, -10, 7, -7, 7, -4, 0], [-2, 0, 0, 6, 10, -10, 0, -4], [-5, 0, 0, 3, 4, 3, 0, -1], [0, 11, -10, -7, 7, -7, 10, 0], [0, -11, -18, 13, -13, 13, -4, 0], [0, -12, 14, 12, -12, 12, -20, 0], [-4, 0, 0, -8, -8, -2, 0, 0], [10, 0, 0, 1, 1, -13, 0, 0], [0, -6, -19, 11, -11, 11, -1, 0], [0, -8, -4, -2, 2, -2, 12, 0], [0, 4, -16, -1, 1, -1, 16, 0], [-15, 0, 0, 4, -3, 4, 0, 7], [2, 0, 0, 14, 6, 2, 0, 8], [-4, 0, 0, 9, -1, 7, 0, 10], [0, 4, -19, 3, -3, 3, -9, 0], [0, -7, 18, 5, -5, 5, -2, 0], [0, 2, 12, -12, 12, -12, 14, 0], [-16, 0, 0, 6, -4, 8, 0, 10], [2, 0, 0, -10, -8, 10, 0, -2], [0, 0, -16, 13, -13, 13, 8, 0], [0, 12, -4, -3, 3, -3, -4, 0], [0, -2, 3, 13, -13, 13, -7, 0], [15, 0, 0, 8, 5, 2, 0, 3], [2, 0, 0, -8, 9, -15, 0, -17], [-12, 0, 0, -4, -6, 16, 0, 2], [-5, 0, 0, -10, -9, 4, 0, -1], [0, -9, -16, 13, -13, 13, -14, 0], [0, -4, 12, -8, 8, -8, -12, 0], [2, 0, 0, 16, 8, 12, 0, 8], [0, 2, -19, -11, 11, -11, 19, 0], [-15, 0, 0, 0, 9, 12, 0, -9], [0, -3, 6, 1, -1, 1, -2, 0], [14, 0, 0, -2, -1, -7, 0, -1], [-18, 0, 0, 0, -1, -11, 0, 1], [0, 11, 6, -1, 1, -1, -10, 0], [0, 6, 25, -16, 16, -16, 9, 0], [0, -4, -16, -14, 14, -14, 8, 0], [16, 0, 0, -14, 4, -6, 0, -18], [-5, 0, 0, -18, -23, 2, 0, 5], [0, 0, 0, 7, 1, -11, 0, 6], [0, 12, 27, -1, 1, -1, -3, 0], [0, -4, 21, 3, -3, 3, -13, 0], [-6, 0, 0, 0, 4, -20, 0, -4], [7, 0, 0, 8, 7, 4, 0, 1], [0, 6, -22, -4, 4, -4, -4, 0], [0, -16, -30, 19, -19, 19, -2, 0], [2, 0, 0, 8, 10, -8, 0, -2], [0, -9, 22, 5, -5, 5, -14, 0], [0, 0, -14, -5, 5, -5, 10, 0], [-34, 0, 0, 3, -5, 1, 0, 8], [0, 1, 16, -5, 5, -5, -18, 0], [0, 8, -9, -21, 21, -21, 29, 0], [0, 8, -1, 6, -6, 6, 3, 0], [1, 0, 0, -8, 11, -6, 0, -19], [-4, 0, 0, -16, -14, 0, 0, -2], [20, 0, 0, -7, -11, 17, 0, 4], [0, -10, -6, 10, -10, 10, 4, 0], [20, 0, 0, -6, -8, 22, 0, 2], [0, 0, 19, -14, 14, -14, 7, 0], [0, -1, -14, -5, 5, -5, 2, 0], [0, 15, -4, -9, 9, -9, 4, 0], [10, 0, 0, 6, -4, -2, 0, 10], [0, 6, 6, -4, 4, -4, 0, 0], [0, 2, 3, -14, 14, -14, 27, 0], [3, 0, 0, 18, 15, 4, 0, 3], [0, 14, -4, 2, -2, 2, -18, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_85_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_85_2_b_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_85_2_b_a(:prec:=8)
    chi := MakeCharacter_85_b();
    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_85_2_b_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_85_2_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_85_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;