
// Make newform 1104.2.a.o in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1104_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1104_2_a_o();".
// 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_1104_2_a_o();".
// 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, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, -2, 2]];
    Rf_basisdens := [1, 1, 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_1104_a();"
function MakeCharacter_1104_a()
    N := 1104;
    order := 1;
    char_gens := [415, 277, 737, 97];
    v := [1, 1, 1, 1];
    // 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;

function MakeCharacter_1104_a_Hecke(Kf)
    return MakeCharacter_1104_a();
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], [-1, 0, 0], [0, 0, 1], [-1, -1, 0], [-1, 1, 1], [2, 0, 0], [1, -1, 0], [-2, 0, 1], [1, 0, 0], [0, 2, 0], [2, 2, 0], [5, 1, -1], [0, 2, 2], [0, -2, -1], [6, 2, 0], [-2, -2, -3], [4, 0, -2], [7, -1, 1], [0, 2, 1], [2, -2, 2], [2, 0, 0], [9, 1, 0], [1, -1, 3], [-3, -1, -2], [6, 0, 2], [-2, 0, -4], [3, -1, 2], [-3, 3, -1], [-3, 5, 1], [-3, -1, 2], [10, 2, 4], [-2, -2, -2], [-5, 1, 0], [4, 0, 0], [0, 0, -3], [4, 4, 0], [1, -3, 3], [0, 0, -2], [-10, 2, -2], [-16, 2, 0], [2, 2, -4], [5, -3, 1], [0, -4, -2], [-8, -2, -2], [-2, 4, -2], [1, -7, -4], [-4, -4, -2], [-12, -4, -4], [-11, -1, 1], [-3, -3, 1], [10, 0, -4], [4, 4, 4], [14, 0, 2], [-17, -3, -1], [4, -2, 2], [6, -2, -4], [-6, 0, 2], [18, 2, 0], [8, 6, -2], [17, 3, 2], [-10, 4, -1], [-2, -2, 1], [4, 4, -6], [-4, 4, 2], [-4, -2, 6], [20, -2, 0], [0, 0, -2], [0, 6, 4], [-6, -6, 0], [4, -2, 2], [8, 6, 0], [12, 0, -2], [-3, -3, -4], [21, 1, -1], [-2, 8, 1], [-18, 6, 4], [-4, 0, 7], [-2, 0, -8], [17, 7, -4], [-16, -2, -6], [5, -5, -1], [15, 3, 3], [2, -10, -6], [-4, -2, -2], [12, 4, 0], [-16, -4, -8], [-16, -6, -2], [14, 0, -2], [-6, -8, -6], [-12, -4, 0], [11, 1, -1], [16, 0, -4], [-4, 4, -4], [6, -2, 0], [-8, 8, 6], [2, 6, -2], [0, 10, 0], [15, 5, -4], [16, 2, 5], [-24, -6, 2], [20, 4, -2], [12, 0, -5], [-27, -1, 1], [-3, 3, 4], [-16, -6, 1], [-12, 2, 6], [10, -6, -6], [20, -2, -2], [32, 0, 0], [28, 2, 6], [24, 0, 0], [7, -1, 1], [-7, -5, -10], [-18, -4, -9], [15, 3, -2], [29, 3, -4], [-16, 6, -1], [8, 0, 6], [-10, 0, 0], [7, -11, -5], [-9, -9, -3], [-4, 2, 6], [-4, -12, 1], [22, 6, 2], [16, 0, 6], [10, -10, -1], [15, 3, 7], [4, -4, 2], [-9, -1, -4], [11, 7, 11], [-8, -4, 8], [-18, -2, 4], [-15, 5, 2], [25, -7, -3], [0, 6, 0], [-32, 2, 2], [18, 6, 3], [12, -6, -5], [-4, 0, -1], [-16, 6, 4], [-16, -8, -2], [14, 8, 8], [10, -6, 0], [-21, -7, -9], [-12, -2, -2], [-2, -2, 0], [38, 0, 0], [18, 0, 0], [-20, 0, 0], [-32, 0, 2], [16, -2, 2], [-39, 3, 2], [0, 0, 2], [-18, -6, 8], [10, 8, 7], [6, -6, -6], [-19, 5, 4], [-4, -2, 6], [-34, 4, 0], [-36, -4, -3], [-2, -10, -8], [5, 3, -8], [10, 10, -4], [25, 11, -7], [31, 5, 0], [-14, -10, -2], [14, -2, 12], [-12, 6, -6], [10, -4, -2], [-18, 8, 4], [23, 5, 15], [-4, 2, -2], [14, -2, -4], [16, 6, -4], [-18, 6, -4], [-25, -7, 2], [-4, -8, -4], [4, 2, 2], [-13, -1, -6], [-15, -3, -1], [-14, 2, 8], [3, 1, 7], [-24, 6, -2], [-24, -10, 8], [-30, -2, -2], [-8, -12, -9], [6, 8, 8], [-32, -2, 3], [-10, -4, -12], [18, -2, -4], [14, 4, 8], [2, -6, -2], [4, -6, -9], [-4, -6, 2], [33, -1, 3], [-33, -7, -6], [-2, -12, 0], [-1, -5, 3], [-1, 1, -10], [0, 0, -12], [-16, 4, 3], [-2, 6, 0], [-4, -14, -6], [-8, -2, -4], [-5, -3, 9], [-16, 2, -4], [15, -1, 12], [-28, 8, -2], [12, 6, 2], [-12, 12, 6], [-14, 0, 4], [-4, -14, -10], [-7, -3, -2], [11, 5, 9], [14, 2, -2], [-8, 2, -14], [-4, -4, -12], [16, -2, -4], [-52, -4, 4], [-16, -10, 2], [38, 4, 4], [-19, -7, -14], [8, 14, 8], [27, -1, 10], [-8, -12, 0], [18, -16, -8], [-11, -1, -6], [10, 6, 0], [1, -3, 10], [-4, 0, 4], [-4, -10, -2], [6, -4, -9], [49, -3, -2], [10, -12, 2], [-10, 4, 7], [-22, -10, 6], [10, -12, -6], [-12, -8, -5], [-14, 2, -10], [6, -14, -4], [-3, -1, -11], [0, 16, 10], [22, -2, 12], [2, 8, 0], [-2, 4, 8], [28, 4, 8], [-26, -2, 0], [-23, -9, -5], [2, -8, -5], [4, 4, 8], [-19, -11, -7], [35, -11, -2], [6, 14, -8], [-4, 10, 8], [-12, 2, 6], [-28, 0, 4], [5, 5, -11], [28, -2, -3], [-26, -4, 2], [48, -2, -2], [-9, 3, 6], [-23, 15, 3], [32, 14, -2], [3, -21, -3], [6, -20, -8], [34, 0, 3], [32, -4, 3], [-5, -3, -6], [-6, 0, -5], [-10, -12, -2], [8, -10, 2], [44, -6, 3], [8, 14, -8], [45, 1, 2], [6, 4, -8], [8, 8, 4], [-16, -4, -10], [-10, -8, 16], [46, 4, -8], [-21, -11, 1], [44, -4, 2], [-25, 3, 10], [-12, 16, 2], [3, 3, -3], [24, 16, 2], [36, 4, -2], [34, -4, 6], [44, 8, 11], [-32, 16, 8], [-14, -12, -2], [-8, -8, 1], [-15, -1, -1], [6, -24, -6], [-7, 3, -11], [20, 14, -6], [0, -8, 1], [27, -5, -4], [-38, 4, 0], [-8, -4, -12], [-4, 8, 8], [-20, 6, -6], [40, -12, -1], [7, -9, -12], [-16, 4, -6], [2, -4, -7], [12, -6, -6], [40, -12, -4], [29, 5, 1], [38, 6, 16], [-26, -4, -4], [24, -8, 2], [10, -6, 1], [-29, 9, 8], [-24, 4, 0], [10, 14, 10], [-24, -10, 0], [24, 12, -4], [14, -22, -12], [-28, -6, 0], [-12, 6, -2], [44, 2, 15], [16, -6, -2], [-8, -2, 14], [0, -16, 4], [1, 23, 12], [22, -8, 6], [-32, -2, -5], [4, -8, 4], [0, 8, 8], [-24, -4, 11], [20, -14, 2], [34, 12, 4], [0, 16, 4], [-2, 22, 14], [26, -8, -9], [18, 2, 0], [-7, -3, -17], [-63, -5, -6], [64, -2, 6], [15, -13, -6], [6, 12, 12], [13, 11, 4], [8, -26, -14], [27, -5, 8], [10, -10, 3], [-16, 4, 8], [-18, -4, -20], [-40, -8, 6], [42, 6, -10], [-30, 2, 1], [4, 0, -20], [24, 6, 6], [-10, 20, 10], [25, -7, -12], [-33, -9, -15], [-32, -2, 8], [14, 14, -16], [-21, -7, 11], [32, 6, -4], [22, 2, -12], [-45, -13, 9], [10, -4, -6], [-30, -2, 2], [-7, 11, -3], [12, -8, -4], [34, 8, 20], [-28, -22, 2], [-15, 1, -16], [-12, -2, 10], [30, 14, -8], [-76, -4, -2], [-46, -10, 6], [54, -6, -1], [-21, 15, 6], [-34, 0, 0], [18, 2, 14], [26, -18, -2], [18, 16, -4], [47, -3, -4], [-12, 10, 18], [16, -4, -5], [-25, -3, 8], [-38, 2, -8], [0, -14, 2], [-30, -4, 5], [8, 0, -4], [2, 10, -16], [24, -10, -10], [-32, -14, -3], [54, 2, 2], [-8, -10, 4], [26, 16, 6], [8, 12, 14], [56, -8, -2], [2, -18, 6], [-2, 16, 6], [-11, 9, -14], [-9, -3, -12], [-36, -6, 15], [-12, 18, 10], [58, 4, 4], [-14, 12, 2], [17, -7, 0], [33, -1, -8], [16, 14, 6], [16, 24, -4], [45, 5, 21], [6, -16, 6], [30, 12, 3], [-30, -14, 0], [40, 6, 10], [-16, -10, -18], [1, -17, -1], [24, -6, 1], [-16, -10, -4], [64, 2, 0], [38, -14, 0], [-2, -10, -4], [-57, -11, 4], [4, 8, 10], [-28, -8, -17], [-7, -3, -9], [52, 4, 8], [-20, 8, 10], [6, -12, -4], [6, 0, 24], [5, 7, 9], [-86, 0, 0], [-8, -24, -10], [26, -26, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1104_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1104_2_a_o();".
// 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_1104_2_a_o(:prec:=3)
    chi := MakeCharacter_1104_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1104_2_a_o();".
// 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_1104_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1104_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -16, 0, 1]>,<7,R![-8, -12, 2, 1]>,<11,R![-32, -16, 4, 1]>],Snew);
    return Vf;
end function;