
// Make newform 1170.2.bs.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1170_bs();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1170_bs_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_1170_2_bs_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_1170_2_bs_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_1170_bs();"
function MakeCharacter_1170_bs()
    N := 1170;
    order := 6;
    char_gens := [911, 937, 1081];
    v := [6, 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_1170_bs_Hecke();"
function MakeCharacter_1170_bs_Hecke(Kf)
    N := 1170;
    order := 6;
    char_gens := [911, 937, 1081];
    char_values := [[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, 0, 1], [0, 0, 0, 0], [0, 0, 0, 1], [0, 3, 0, 0], [2, -2, -1, 2], [0, 1, 0, -4], [4, 0, -4, 0], [1, 4, 1, 0], [0, 2, 0, 2], [0, 2, -2, 2], [4, 0, -8, 2], [-8, 1, 4, -1], [0, 4, 0, -4], [-6, 0, 6, 0], [2, 0, -4, -3], [2, -2, 0, 1], [2, 8, 2, 0], [4, -2, -4, 4], [4, -2, -2, 2], [4, 6, 4, 0], [-4, 0, 8, 0], [10, -8, 0, 4], [-2, 0, 4, 6], [14, 2, -7, -2], [6, -2, 6, 0], [0, 6, 2, 6], [4, -18, 0, 9], [0, -4, -6, -4], [-6, 0, 12, 0], [-12, 0, 12, 0], [0, 5, 4, 5], [3, -20, 0, 10], [-2, -10, -2, 0], [-1, 12, 1, -24], [-12, 2, -12, 0], [12, 0, -24, -2], [16, 6, 0, -3], [-4, 4, -4, 0], [4, -3, -2, 3], [10, -7, -10, 14], [0, -4, 16, -4], [10, 8, 0, -4], [0, -10, 0, 20], [-16, 4, 8, -4], [12, 1, -6, -1], [18, 4, -18, -8], [0, -4, 15, -4], [-16, -5, 8, 5], [2, -10, 2, 0], [-2, 0, 4, -8], [-18, 0, 0, 0], [2, 0, -4, 0], [-3, -20, -3, 0], [23, -2, -23, 4], [0, -2, 18, -2], [0, -13, -2, -13], [-24, 4, 24, -8], [12, 14, -6, -14], [4, -9, -4, 18], [4, 0, -8, 0], [0, 6, -4, 6], [-14, -5, -14, 0], [14, 0, -28, -4], [-12, -8, 0, 4], [-16, -20, 0, 10], [2, 0, -4, -27], [6, 4, 6, 0], [-16, -12, 0, 6], [-6, 6, 6, -12], [4, -14, -2, 14], [12, -18, -6, 18], [4, 0, -8, 6], [0, -10, -4, -10], [-16, 4, 16, -8], [-2, -8, 1, 8], [-12, 16, -12, 0], [-4, -16, 0, 8], [-4, 1, -4, 0], [14, 10, -7, -10], [-11, -20, -11, 0], [0, 8, -4, 8], [-8, 0, 16, 8], [16, 0, -8, 0], [-12, 12, 12, -24], [0, 10, -4, 10], [6, 16, 0, -8], [7, -6, 7, 0], [-4, -4, 2, 4], [-6, 22, -6, 0], [12, 0, -24, -20], [-14, -12, 0, 6], [28, -2, -14, 2], [0, 21, 0, 0], [0, 6, -5, 6], [-10, 0, 20, 16], [2, 1, -2, -2], [-20, -12, 10, 12], [17, -12, 0, 6], [0, 16, 14, 16], [-12, 0, 24, -6], [12, -20, 0, 10], [-12, 33, 6, -33], [2, -10, -2, 20], [0, -10, 1, -10], [-3, 16, 0, -8], [10, 0, -20, 8], [8, 2, -4, -2], [-12, 0, 24, 24], [-22, -8, 0, 4], [0, -16, -3, -16], [-8, 17, 8, -34], [0, 45, 0, -45], [2, 32, 2, 0], [-13, 0, 26, -20], [0, -8, 0, 0], [11, 2, -11, -4], [12, -8, 12, 0], [0, -1, -34, -1], [0, -9, 14, -9], [-12, -16, 12, 32], [12, -32, -6, 32], [0, 12, -4, 12], [-4, -8, 0, 4], [-10, -24, -10, 0], [22, -4, -11, 4], [-6, 12, 0, -6], [-10, -10, -10, 0], [16, 0, -16, 0], [4, -10, 0, 5], [-8, 0, 16, 7], [46, -4, -23, 4], [8, 4, -4, -4], [0, 18, -10, 18], [0, -1, -20, -1], [-15, 18, -15, 0], [32, -12, -16, 12], [-6, 43, -6, 0], [2, -2, 2, 0], [-24, 8, 24, -16], [0, -12, 14, -12], [-9, 0, 18, 8], [-20, 32, 10, -32], [24, -9, -24, 18], [28, 0, -56, 2], [0, -10, 22, -10], [-18, 12, -18, 0], [20, 0, -40, 10], [22, 8, 0, -4], [-13, 8, 0, -4], [4, 0, -8, 4], [16, -6, 16, 0], [0, 22, 1, 22], [4, -20, 0, 10], [0, 1, 54, 1], [0, -4, -14, -4], [8, -24, 0, 12], [-4, 30, 4, -60], [-8, -44, -8, 0], [6, -32, 0, 16], [-28, 0, 56, 12], [-12, -24, 6, 24], [-36, -10, 36, 20], [-8, 0, 16, -15], [-31, -6, 31, 12], [4, 2, -2, -2], [18, 0, -36, -15], [0, -18, -8, -18], [-36, -15, 36, 30], [-19, 0, 38, 20], [-12, -48, 0, 24], [-10, 0, 20, 28], [-12, -26, -12, 0], [6, 16, -6, -32], [6, 48, 6, 0], [-6, -32, 0, 16], [-3, 22, 3, -44], [-12, 40, 6, -40], [-4, 0, 8, -48], [0, -2, 0, -2], [0, -18, 8, -18], [8, 0, -16, 28], [7, -4, 0, 2], [16, -22, 0, 11], [32, 0, -64, -4], [20, 15, -10, -15], [18, 12, -18, -24], [-20, -2, 0, 1], [-14, 0, 28, -26], [-10, -32, 5, 32], [-4, -18, -4, 0], [-2, -32, 2, 64], [18, -14, 18, 0], [-39, -16, 0, 8], [8, -42, -4, 42], [-12, -2, 12, 4], [0, 4, 6, 4], [0, 0, 0, 52], [44, 13, -44, -26], [4, 0, -8, -12], [-8, -36, 0, 18], [-2, -12, -2, 0], [-16, 14, 16, -28], [0, -26, 0, 26], [21, -16, 0, 8], [-30, -22, 15, 22], [0, -8, 24, -8], [0, 0, 0, 56], [-34, 2, 34, -4], [50, -10, -25, 10], [-49, -4, 49, 8], [0, -14, 38, -14], [12, -36, 0, 18], [0, 1, -40, 1], [0, 34, 0, 0], [-4, 18, -4, 0], [-25, 0, 50, 24], [-12, -4, 0, 2], [13, 18, -13, -36], [4, 37, -2, -37], [14, 0, -28, 33], [0, -14, 22, -14], [24, 0, -48, -26], [15, 0, -30, -22], [-16, -20, -16, 0], [0, -10, 36, -10], [-30, 4, 0, -2], [0, 2, -60, 2], [-10, 0, 10, 0], [28, -14, -28, 28], [-9, 0, 18, -22], [0, -13, 12, -13], [0, 0, -41, 0], [-36, -42, 18, 42], [5, 4, 0, -2], [-4, -32, 0, 16], [-10, 0, 20, 35], [0, 32, 0, 0], [-40, 30, 20, -30], [-17, 10, 17, -20], [0, 12, -34, 12], [-32, -2, 16, 2], [0, -12, 16, -12], [20, -13, -20, 26], [24, -34, -12, 34], [8, 42, 8, 0], [-18, 52, 0, -26], [16, -16, 16, 0], [-10, 38, 5, -38], [21, -28, 21, 0], [0, -3, 18, -3], [-24, -18, 12, 18], [-22, 18, 11, -18], [-34, 0, 68, -9], [0, -12, -26, -12], [46, 6, 0, -3], [7, 54, 7, 0], [-20, -6, 20, 12], [-36, 16, 18, -16], [30, -6, 0, 3], [-24, -2, -24, 0], [4, 50, 0, -25], [0, 16, 44, 16], [12, 0, -24, -28], [0, 4, -48, 4], [12, -6, 12, 0], [8, 8, -8, -16], [-2, -28, -2, 0], [11, 0, -22, 2], [-14, 2, -14, 0], [48, -8, -48, 16], [42, 32, 0, -16], [-22, 0, 44, 34], [20, -14, -10, 14], [-22, 14, 22, -28], [56, 12, -56, -24], [40, 20, -20, -20], [4, -42, 4, 0], [38, 0, -76, -8], [16, 20, 16, 0], [4, -8, -4, 16], [0, -5, -62, -5], [-12, 26, 6, -26], [-16, -12, 0, 6], [-24, 28, 12, -28], [-6, 0, 12, 28], [-36, 12, 0, -6], [18, -64, 0, 32], [-34, 0, 68, -13], [2, 24, 2, 0], [-67, 2, 67, -4], [0, -12, -28, -12], [12, 4, -12, -8], [-24, -32, 12, 32], [17, 38, 17, 0], [-8, -32, 8, 64], [-10, -72, 0, 36], [-2, 16, 0, -8], [0, 11, -46, 11], [0, 36, -8, 36], [-20, 26, 10, -26], [-44, -4, 44, 8], [20, 0, -40, -14], [0, 16, 0, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1170_bs_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1170_2_bs_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_1170_2_bs_d(:prec:=4)
    chi := MakeCharacter_1170_bs();
    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_1170_2_bs_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_1170_2_bs_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1170_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![81, 0, -9, 0, 1]>,<11,R![1, 6, 11, -6, 1]>],Snew);
    return Vf;
end function;