
// Make newform 1170.2.e.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1170_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1170_e_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_e_f();".
// 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_e_f();".
// 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 := [40, -12, 1, 16, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [240, -966, 120, 183, -70, 4], [-1000, 643, 390, -184, -5, 13], [-180, 57, -1, -26, 8, -3], [-6, -7, -92, 11, 24, -9]];
    Rf_basisdens := [1, 1, 445, 445, 89, 178];
    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_1170_e();"
function MakeCharacter_1170_e()
    N := 1170;
    order := 2;
    char_gens := [911, 937, 1081];
    v := [2, 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_1170_e_Hecke();"
function MakeCharacter_1170_e_Hecke(Kf)
    N := 1170;
    order := 2;
    char_gens := [911, 937, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 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+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, 0, 1], [0, 0, 0, 0, 0, 0], [0, -1, -1, 1, -1, 1], [0, 1, 1, 0, -1, 0], [-2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 2, 2, -1, -1, 0], [-2, 2, -2, -2, 0, 0], [0, 0, 0, -2, 2, -2], [-2, 2, -2, -2, 0, 0], [4, -2, -2, 0, -2, 0], [0, 1, 1, -2, 1, 2], [-2, 0, -4, -2, -2, 0], [0, 0, 0, 3, -3, -4], [0, 1, 1, 0, -1, -4], [0, 0, 0, -2, 2, -2], [2, -2, 2, 2, 0, 0], [2, 0, -4, -2, -2, 0], [0, 0, 0, -2, 2, 4], [4, -1, 3, 2, 1, 0], [0, 0, 0, 0, 0, -6], [-8, -2, 2, 2, 0, 0], [0, 2, 2, -2, 0, 4], [-2, 2, -2, -2, 0, 0], [0, -2, -2, 6, -4, 10], [6, -8, -4, 2, -6, 0], [0, 4, 4, -2, -2, -2], [0, 0, 0, -2, 2, 0], [-8, 3, 3, 0, 3, 0], [0, 0, 0, 0, 0, -4], [0, 0, 0, -2, 2, -6], [4, 4, 6, 1, 5, 0], [0, 0, 0, -2, 2, 10], [-4, 4, 2, -1, 3, 0], [0, 0, 0, 0, 0, 0], [4, 3, 7, 2, 5, 0], [0, -2, -2, 2, 0, -6], [0, 2, 2, -6, 4, -4], [0, 0, 0, 8, -8, 0], [0, 0, 0, 2, -2, 14], [-4, 4, 2, -1, 3, 0], [-2, -6, -6, 0, -6, 0], [-8, 6, -2, -4, 2, 0], [0, 0, 0, 6, -6, -2], [0, -1, -1, 6, -5, 14], [0, -10, -2, 4, -6, 0], [8, 4, -2, -3, 1, 0], [0, -5, -5, 12, -7, 12], [0, -4, -4, 0, 4, -4], [-12, 7, -1, -4, 3, 0], [0, 2, 2, 7, -9, 0], [8, -3, 1, 2, -1, 0], [2, 10, -2, -6, 4, 0], [12, -4, -4, 0, -4, 0], [0, 2, 2, 1, -3, 8], [0, 2, 2, 0, -2, 2], [-6, -4, 0, 2, -2, 0], [-4, -9, -5, 2, -7, 0], [0, -6, -6, 8, -2, 6], [-14, 2, 6, 2, 4, 0], [0, 0, 0, -10, 10, 4], [0, -3, -3, -2, 5, -6], [0, -2, -2, 4, -2, 12], [-16, 4, -8, -6, -2, 0], [0, -2, -2, 3, -1, 20], [0, -4, -4, 4, 0, 2], [10, 2, 10, 4, 6, 0], [0, 2, 2, 5, -7, -12], [0, 4, 4, -5, 1, -20], [-16, 11, 3, -4, 7, 0], [0, 0, 0, 10, -10, -14], [12, -2, 6, 4, 2, 0], [0, 4, 4, -4, 0, -22], [0, 6, 6, -8, 2, -10], [-2, 2, -6, -4, -2, 0], [0, -5, -5, 8, -3, 8], [-10, 0, -16, -8, -8, 0], [0, 8, 8, -4, -4, 10], [-6, -6, 2, 4, -2, 0], [10, -10, 6, 8, -2, 0], [-4, 4, 10, 3, 7, 0], [0, -13, 3, 8, -5, 0], [4, -1, -5, -2, -3, 0], [0, 2, 2, -3, 1, -12], [-8, 8, 4, -2, 6, 0], [0, 8, 8, -1, -7, -4], [6, -16, 0, 8, -8, 0], [0, 4, 4, 0, -4, -6], [-8, -9, -1, 4, -5, 0], [0, -8, -8, 4, 4, -4], [0, 0, 0, 2, -2, -24], [-16, -9, 3, 6, -3, 0], [0, 0, 0, 0, 0, -8], [4, 4, 10, 3, 7, 0], [-6, -4, 8, 6, 2, 0], [0, 2, 2, -6, 4, -2], [0, 0, -8, -4, -4, 0], [30, -2, -4, -1, -3, 0], [0, 12, 12, -10, -2, 0], [20, -3, -3, 0, -3, 0], [0, -4, -4, 3, 1, -8], [0, -1, -1, -6, 7, -14], [0, -4, -4, 1, 3, 28], [-14, -6, -4, 1, -5, 0], [0, 12, 6, -3, 9, 0], [0, 0, 0, -8, 8, -10], [0, 2, 2, 6, -8, -12], [0, 2, 2, -6, 4, 18], [8, 12, 16, 2, 14, 0], [2, 10, 4, -3, 7, 0], [0, 6, 6, 2, -8, -2], [0, 8, 8, -16, 8, -6], [0, -2, -2, -4, 6, 6], [30, 0, 12, 6, 6, 0], [-20, 7, -5, -6, 1, 0], [18, 0, -8, -4, -4, 0], [0, -4, -4, -6, 10, -12], [0, 2, 2, -10, 8, -6], [0, 2, 2, 0, -2, 2], [8, 12, -4, -8, 4, 0], [8, 4, 12, 4, 8, 0], [0, 2, 2, 5, -7, 0], [0, -8, -8, 2, 6, -18], [0, 2, 2, -2, 0, 4], [-6, -12, 0, 6, -6, 0], [6, 2, -14, -8, -6, 0], [16, 4, 12, 4, 8, 0], [-16, 6, -2, -4, 2, 0], [0, -4, -4, 0, 4, -30], [0, 5, 5, 10, -15, -6], [10, 0, 12, 6, 6, 0], [0, 13, 13, -12, -1, -8], [0, -8, -16, -4, -12, 0], [0, -4, -4, 2, 2, -2], [-14, 2, 6, 2, 4, 0], [-18, 8, 4, -2, 6, 0], [0, 3, 3, -2, -1, 22], [0, -8, -8, -4, 12, 12], [0, -6, -6, 18, -12, 2], [10, -10, -16, -3, -13, 0], [22, 0, 20, 10, 10, 0], [-24, -3, -11, -4, -7, 0], [0, -2, -2, 18, -16, 10], [0, -10, -10, 16, -6, 12], [22, -2, -6, -2, -4, 0], [8, 2, -14, -8, -6, 0], [0, 11, 11, -2, -9, -6], [0, 12, 12, -8, -4, -8], [36, 4, -4, -4, 0, 0], [0, -5, -5, 4, 1, -28], [0, 1, 1, 10, -11, 10], [18, -10, 4, 7, -3, 0], [0, -8, -8, 15, -7, 20], [0, -10, -10, 16, -6, 2], [0, 16, 16, -9, -7, 4], [-24, 2, 10, 4, 6, 0], [-24, 4, -12, -8, -4, 0], [-10, 20, 16, -2, 18, 0], [0, 4, 4, -16, 12, 0], [12, -9, 15, 12, 3, 0], [0, 4, 4, 2, -6, -12], [0, -2, -2, 17, -15, -12], [0, 5, 5, 4, -9, -32], [-16, -16, 2, 9, -7, 0], [0, -10, -10, -2, 12, 10], [0, 3, 3, 4, -7, -24], [8, -16, -12, 2, -14, 0], [0, 0, 0, 4, -4, -22], [-2, 2, 6, 2, 4, 0], [0, 0, 0, 8, -8, -2], [-26, 16, 8, -4, 12, 0], [-20, 7, 7, 0, 7, 0], [0, 8, 0, -4, 4, 0], [0, -2, -2, 4, -2, 14], [-8, 18, 2, -8, 10, 0], [30, -10, 8, 9, -1, 0], [-18, 0, 4, 2, 2, 0], [0, 8, -8, -8, 0, 0], [0, -2, -2, 0, 2, -2], [6, 14, -10, -12, 2, 0], [0, -8, -8, 24, -16, 24], [-40, 8, 0, -4, 4, 0], [0, 0, 0, 4, -4, -30], [0, -4, -4, 4, 0, 22], [0, 0, 0, 0, 0, -4], [2, 8, 8, 0, 8, 0], [0, 6, 6, 6, -12, -6], [0, -6, -6, -12, 18, 28], [2, 8, 12, 2, 10, 0], [12, -14, 2, 8, -6, 0], [0, -10, -10, 17, -7, -8], [0, 4, 4, 0, -4, -12], [-8, 0, 6, 3, 3, 0], [16, 12, 4, -4, 8, 0], [0, 8, 8, 3, -11, -16], [0, -8, -8, 0, 8, 12], [6, 16, -12, -14, 2, 0], [0, -4, -4, 2, 2, 30], [0, 0, 0, -2, 2, 30], [0, 2, 2, 2, -4, 30], [16, 9, 9, 0, 9, 0], [-8, -10, -6, 2, -8, 0], [0, 1, 1, -14, 13, 34], [2, -22, -8, 7, -15, 0], [-14, 24, 4, -10, 14, 0], [0, 2, 2, 2, -4, 22], [24, -9, -5, 2, -7, 0], [0, 4, 4, -14, 10, -8], [22, -12, 4, 8, -4, 0], [-20, 12, 2, -5, 7, 0], [0, 0, 0, -4, 4, 56], [6, 2, 6, 2, 4, 0], [0, 8, 8, -2, -6, 10], [0, -18, -18, 20, -2, 4], [16, 9, 5, -2, 7, 0], [-22, 14, 14, 0, 14, 0], [0, 2, 2, -8, 6, -34], [-26, 14, 4, -5, 9, 0], [0, 0, 0, -8, 8, -4], [0, 11, 11, 10, -21, -26], [2, 20, 8, -6, 14, 0], [8, -3, -15, -6, -9, 0], [10, -20, -8, 6, -14, 0], [0, -1, -1, 4, -3, 12], [0, 0, 0, 7, -7, 16], [14, -4, 20, 12, 8, 0], [0, -10, -10, -5, 15, -4], [0, 30, 2, -14, 16, 0], [0, -10, -10, 18, -8, -6], [6, -8, 8, 8, 0, 0], [0, 6, 6, -14, 8, -34], [36, 4, -4, -4, 0, 0], [-40, 2, -14, -8, -6, 0], [-14, 22, 20, -1, 21, 0], [0, 12, 12, -1, -11, 16], [0, -13, -13, 8, 5, -4], [34, 6, 10, 2, 8, 0], [0, 5, 5, -2, -3, -42], [8, 12, 20, 4, 16, 0], [40, -22, -6, 8, -14, 0], [0, -2, -2, 0, 2, 20], [12, 0, 18, 9, 9, 0], [0, 6, 6, 0, -6, -18], [-36, -9, -17, -4, -13, 0], [0, 6, 6, 14, -20, -10], [16, 14, -6, -10, 4, 0], [0, 5, 5, -4, -1, 36], [-14, -8, 12, 10, 2, 0], [-2, -8, 0, 4, -4, 0], [0, 10, 10, -2, -8, -22], [0, -9, -9, 2, 7, -2], [38, 20, -4, -12, 8, 0], [0, 5, 5, -16, 11, 24], [14, -14, -4, 5, -9, 0], [0, 0, 0, -18, 18, -26], [34, -4, 4, 4, 0, 0], [-30, 22, 6, -8, 14, 0], [0, 10, 10, -8, -2, -20], [0, 4, 4, -10, 6, 10], [0, -12, -12, 16, -4, 30], [0, 10, 10, -24, 14, -22], [0, 16, 16, -27, 11, -32], [-12, -13, -5, 4, -9, 0], [0, -20, -20, 12, 8, 2], [0, 10, 10, -16, 6, 14], [-16, -4, -10, -3, -7, 0], [-8, -7, -15, -4, -11, 0], [-2, 30, 14, -8, 22, 0], [0, 12, 12, -8, -4, -12], [0, 14, 14, -20, 6, -22], [10, -18, -2, 8, -10, 0], [0, 10, 10, 0, -10, -28], [0, 10, 10, 2, -12, 22], [-8, 18, -10, -14, 4, 0], [0, -4, -4, -8, 12, -12], [0, 4, 4, 0, -4, -44], [0, 6, 6, -16, 10, -28], [-16, -24, 0, 12, -12, 0], [22, -20, 0, 10, -10, 0], [-8, 16, -10, -13, 3, 0], [0, 8, 8, -6, -2, 26], [0, 17, 5, -6, 11, 0], [0, -4, -4, 12, -8, -6], [32, 0, 0, 0, 0, 0], [0, -2, -2, 10, -8, 36], [16, -8, 12, 10, 2, 0], [0, 10, 10, -13, 3, -48], [0, 17, 17, -2, -15, -14], [-8, 13, 1, -6, 7, 0], [-14, -16, 8, 12, -4, 0], [6, -8, 4, 6, -2, 0], [0, 0, 0, 3, -3, -60], [0, 10, 10, -12, 2, -42], [-30, 8, 8, 0, 8, 0], [0, 0, 0, -8, 8, 2], [-50, 20, 16, -2, 18, 0], [32, -10, -18, -4, -14, 0], [0, 6, 6, -28, 22, 2], [-16, -4, 14, 9, 5, 0], [0, 0, 0, -10, 10, -28], [0, -14, -14, 7, 7, 20], [0, -9, -9, -18, 27, 34], [-32, 22, 22, 0, 22, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1170_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1170_2_e_f();".
// 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_e_f(:prec:=6)
    chi := MakeCharacter_1170_e();
    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_e_f();".
// 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_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1170_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![400, 0, 305, 0, 34, 0, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;