
// Make newform 1170.2.bs.e in Magma, downloaded from the LMFDB on 24 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_e();".
// 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_e();".
// 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, 2, 0, 0], [-4, 3, 2, -3], [-1, 0, 4, 0], [-4, 0, 4, 0], [2, -4, 2, 0], [0, -1, 2, -1], [0, 1, 2, 1], [-1, 0, 2, 0], [6, -4, -3, 4], [0, 2, 0, -2], [-5, 4, 5, -8], [-2, 0, 4, 7], [6, 8, 0, -4], [-2, 5, -2, 0], [0, -6, 0, 12], [-4, 8, 2, -8], [6, -2, 6, 0], [0, 0, 0, 2], [-7, 8, 0, -4], [-4, 0, 8, -2], [-4, -4, 2, 4], [-2, -4, -2, 0], [0, -4, -4, -4], [2, 16, 0, -8], [0, 8, 6, 8], [-8, 0, 16, -2], [6, 3, -6, -6], [0, -4, -2, -4], [10, -10, 0, 5], [2, 1, 2, 0], [6, -4, -6, 8], [-2, -17, -2, 0], [-6, 0, 12, 0], [-5, 0, 0, 0], [11, 4, 11, 0], [-20, -1, 10, 1], [4, -4, -4, 8], [0, 1, -18, 1], [4, 8, 0, -4], [18, -2, -18, 4], [20, 6, -10, -6], [-12, -6, 6, 6], [12, 4, -12, -8], [0, -10, -6, -10], [4, 24, -2, -24], [6, -6, 6, 0], [-8, 0, 16, 6], [6, 14, 0, -7], [6, 0, -12, 6], [1, -16, 1, 0], [14, -1, -14, 2], [0, -5, -14, -5], [0, -7, -6, -7], [12, 0, -12, 0], [6, 4, -3, -4], [-3, -4, 3, 8], [4, 0, -8, 2], [0, -4, 3, -4], [8, -18, 8, 0], [2, 0, -4, 16], [18, -12, 0, 6], [-28, 0, 0, 0], [2, 0, -4, -18], [-12, -4, -12, 0], [-8, -20, 0, 10], [12, 6, -12, -12], [4, -18, -2, 18], [0, 2, 0, -2], [4, 0, -8, -12], [0, -6, -26, -6], [-5, -12, 5, 24], [16, -14, -8, 14], [-6, 15, -6, 0], [22, -2, 0, 1], [7, 0, 7, 0], [0, -32, 0, 32], [0, -4, 0, 0], [0, -6, 12, -6], [6, 0, -12, -6], [-12, 18, 6, -18], [-2, 10, 2, -20], [0, -8, 4, -8], [-6, 12, 0, -6], [16, 12, 16, 0], [-4, -28, 2, 28], [-2, -3, -2, 0], [-4, 0, 8, 14], [2, -16, 0, 8], [-4, 2, 2, -2], [18, 8, 18, 0], [0, 10, 0, 10], [-2, 0, 4, 10], [0, 18, 0, -36], [-12, -9, 6, 9], [0, -20, 0, 10], [0, 12, 9, 12], [4, 0, -8, 20], [16, -8, 0, 4], [-32, -10, 16, 10], [-22, -10, 22, 20], [0, -10, 12, -10], [14, 36, 0, -18], [0, 0, 0, -10], [12, 8, -6, -8], [-22, 0, 44, -7], [0, -12, 0, 6], [0, -12, -1, -12], [-12, -18, 12, 36], [-22, 20, 11, -20], [2, -19, 2, 0], [-14, 0, 28, 0], [2, -28, 2, 0], [14, 8, -14, -16], [2, 24, 2, 0], [0, 10, 4, 10], [0, -14, -20, -14], [-2, -1, 2, 2], [20, 26, -10, -26], [0, 0, -32, 0], [-22, 12, 0, -6], [12, -20, 12, 0], [-36, -4, 18, 4], [2, 2, 0, -1], [-4, 16, -4, 0], [0, 20, 0, -40], [-4, 32, 0, -16], [-4, 0, 8, -44], [-20, 2, 10, -2], [4, -37, -2, 37], [0, -4, -21, -4], [0, 0, 18, 0], [4, 12, 4, 0], [-18, -4, 9, 4], [-16, 0, -16, 0], [1, 0, 1, 0], [-24, -8, 24, 16], [0, 12, -22, 12], [12, 0, -24, 20], [12, 25, -6, -25], [-8, 18, 8, -36], [-10, 0, 20, 0], [0, -2, -20, -2], [24, -2, 24, 0], [23, 0, -46, 4], [-2, 26, 0, -13], [-4, 36, 0, -18], [2, 0, -4, -39], [7, 16, 7, 0], [0, 0, -14, 0], [23, -8, 0, 4], [0, -7, 2, -7], [0, -8, 7, -8], [0, 28, 0, -14], [20, 20, -20, -40], [-8, -20, -8, 0], [10, 40, 0, -20], [12, 0, -24, -6], [-4, 6, 2, -6], [-14, -1, 14, 2], [-2, 0, 4, -38], [8, 2, -8, -4], [12, 23, -6, -23], [-6, 0, 12, 47], [0, -8, 39, -8], [6, 24, -6, -48], [3, 0, -6, -12], [-18, 8, 0, -4], [24, 0, -48, 12], [10, 2, 10, 0], [46, 6, -46, -12], [6, -28, 6, 0], [-21, 0, 0, 0], [-38, 2, 38, -4], [0, -18, 0, 18], [20, 0, -40, -10], [0, 0, 0, 0], [0, 16, 10, 16], [2, 0, -4, 2], [-6, 10, 0, -5], [-17, -40, 0, 20], [14, 0, -28, 27], [-4, 39, 2, -39], [10, 13, -10, -26], [43, -8, 0, 4], [13, 0, -26, -4], [2, 28, -1, -28], [24, -12, 24, 0], [-8, -34, 8, 68], [10, 48, 10, 0], [-14, 52, 0, -26], [-4, 27, 2, -27], [16, -24, -16, 48], [0, 8, -12, 8], [17, 0, -34, 16], [-23, -24, 23, 48], [18, 0, -36, 31], [-38, 2, 0, -1], [12, 14, 12, 0], [29, 4, -29, -8], [8, -28, -4, 28], [22, 40, 0, -20], [16, 36, -8, -36], [0, 10, 34, 10], [6, 0, -12, -4], [4, -30, -4, 60], [-40, 22, 20, -22], [8, -22, -8, 44], [0, 6, -34, 6], [-18, 22, 0, -11], [0, 2, 28, 2], [-4, 16, -4, 0], [-30, 12, -30, 0], [-15, 0, 30, -8], [10, 8, 0, -4], [-20, 6, 20, -12], [-8, -8, 4, 8], [-2, 0, 4, 24], [0, -20, -4, -20], [10, 0, -20, 8], [-10, 0, 20, 24], [20, -20, 20, 0], [0, 12, -2, 12], [-6, -24, 0, 12], [0, -27, 2, -27], [12, 32, -12, -64], [16, 18, -16, -36], [-10, 0, 20, 27], [0, 20, 7, 20], [0, 20, -38, 20], [12, 20, -6, -20], [6, 12, 0, -6], [9, 0, 0, 0], [0, 0, 0, -24], [7, 20, 7, 0], [60, -8, -30, 8], [12, -30, -12, 60], [0, 16, 16, 16], [0, -56, 0, 56], [0, -10, 16, -10], [-6, -30, 6, 60], [-60, 4, 30, -4], [24, -14, 24, 0], [14, -32, 0, 16], [-16, -20, -16, 0], [8, 8, -4, -8], [2, -54, 2, 0], [0, 17, 34, 17], [50, 24, -25, -24], [-32, 4, 16, -4], [2, 0, -4, -73], [0, -4, 37, -4], [34, 12, 0, -6], [-12, -20, -12, 0], [4, 24, -4, -48], [12, -40, -6, 40], [54, 0, 0, 0], [8, 10, 8, 0], [8, 4, 0, -2], [0, 6, 56, 6], [-22, 0, 44, 2], [0, 8, 14, 8], [26, -29, 26, 0], [-38, 20, 38, -40], [-28, -14, -28, 0], [-14, 0, 28, 12], [-1, -52, -1, 0], [10, -26, -10, 52], [-8, -16, 0, 8], [-27, 0, 54, -24], [56, 28, -28, -28], [53, 12, -53, -24], [-8, 12, 8, -24], [0, 52, 0, -52], [8, 18, 8, 0], [-14, 0, 28, -48], [-15, 0, -15, 0], [-38, -13, 38, 26], [0, -10, -8, -10], [50, 40, -25, -40], [-6, -46, 0, 23], [40, 4, -20, -4], [3, 0, -6, 52], [-4, 40, 0, -20], [10, 12, 0, -6], [-14, 0, 28, -26], [37, 4, 37, 0], [4, 16, -4, -32], [0, -19, 42, -19], [10, 4, -10, -8], [-4, 27, 2, -27], [-6, -47, -6, 0], [34, -12, -34, 24], [-18, 38, 0, -19], [-11, -64, 0, 32], [0, -22, -44, -22], [0, -29, 6, -29], [70, -28, -35, 28], [50, -4, -50, 8], [-10, 0, 20, 66], [0, -16, -19, -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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1170_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 0, -4, 0, 1]>,<11,R![9, 36, 51, 12, 1]>],Snew);
    return Vf;
end function;