
// Make newform 1805.2.a.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1805_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_1805_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_1805_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 := [3, 4, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -4, 0, 1]];
    Rf_basisdens := [1, 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_1805_a();"
function MakeCharacter_1805_a()
    N := 1805;
    order := 1;
    char_gens := [362, 1446];
    v := [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_1805_a_Hecke(Kf)
    return MakeCharacter_1805_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, -1, 0], [1, -1, 0, 0], [1, 0, 0, 0], [-1, 0, 0, -1], [0, -2, 0, -1], [2, -1, 0, -1], [0, -2, -1, 0], [0, 0, 0, 0], [0, 1, -1, -1], [0, 1, 2, -2], [-1, 1, -3, -1], [-1, 0, -2, 1], [-3, 2, -2, -3], [-1, 4, -1, -1], [-3, 2, 2, 3], [0, -2, 3, -2], [0, -5, 0, 0], [-1, 0, -4, -2], [2, -2, 2, 0], [6, -1, 3, 0], [-4, -3, 1, -1], [5, -1, 2, 1], [0, 2, 1, -2], [3, -3, -2, -3], [-1, 5, 0, -4], [-3, -1, -1, -3], [2, 3, 1, 1], [6, 1, 5, 1], [8, 1, -1, -2], [-3, -3, -2, -2], [8, 2, -2, -2], [-3, 0, -8, 1], [-3, 4, 1, 7], [-4, -5, 4, 1], [-3, 0, 7, 4], [5, 4, -4, 4], [-1, 1, -1, -2], [2, -4, -1, 6], [-3, 6, 0, 3], [-3, 6, -4, 7], [-3, -4, -5, 4], [-10, 7, 2, 1], [3, -1, -4, -5], [8, 4, -3, 2], [-15, -2, -6, 3], [2, 3, -3, -4], [11, -1, 2, 7], [11, -1, 6, -6], [18, -2, -4, -2], [-4, 4, 2, -1], [9, -5, -1, 6], [-9, 9, 3, -1], [5, -3, 4, 7], [0, 4, -8, -1], [-3, -1, -2, 6], [0, -2, 0, -4], [-3, 6, -6, 4], [-13, 0, -5, 6], [8, 2, -2, 6], [-9, -7, -3, 6], [5, -9, 5, -4], [-12, -5, 3, -5], [11, -3, 4, 4], [-9, -2, -2, 1], [14, -1, 7, 1], [-3, -2, -3, -9], [5, -13, 0, 2], [-4, -14, -2, 4], [-3, -2, 5, -3], [5, 3, 9, -6], [0, 4, -5, -6], [0, 4, 5, -6], [8, 3, 6, -7], [11, -6, -9, -5], [8, 5, -3, 2], [-3, -2, -3, 1], [12, 8, 0, 5], [-13, 8, 0, 3], [6, -9, 3, -6], [-13, 0, -1, 2], [3, 1, -5, 3], [-4, 14, 2, -7], [-18, -3, -4, -4], [-7, -1, 4, 2], [-10, 6, -4, 1], [-15, -4, 0, 5], [-9, 4, 7, 2], [-10, 4, 5, -4], [-6, -3, 4, -2], [20, -11, -7, -3], [-18, 9, 0, -9], [6, 12, 7, -3], [14, 12, -5, 0], [3, 0, -5, -4], [-13, -4, -7, 9], [-9, -7, 8, -2], [-18, 0, 1, 6], [-12, 2, -1, 5], [5, -7, -10, 2], [32, -3, 5, 6], [8, 2, 5, -6], [15, -6, 3, -11], [-6, -13, -2, 1], [3, -13, 2, 1], [-7, 2, 6, 2], [2, -10, -1, -8], [-9, 9, -1, 0], [-6, -4, -2, -10], [6, -5, -8, -1], [5, 3, -8, 1], [14, -1, 7, -7], [20, -13, -5, -3], [-6, 3, -5, -1], [-19, 9, -2, 0], [17, -9, 12, 9], [-9, -3, 5, -9], [-7, 4, -10, 9], [9, 4, -11, -1], [3, 1, -2, 0], [-18, 9, 11, 3], [-10, -1, 2, -6], [23, -3, 10, 10], [-3, 6, -5, 9], [9, -9, 10, -2], [14, -9, 2, -12], [3, 3, 2, -2], [2, 7, 0, 14], [-3, -6, 8, 0], [-7, -2, 5, -5], [-7, 5, -2, 10], [-1, 17, -1, 1], [3, -4, -1, -1], [-4, 11, -4, 0], [-10, 3, 0, -9], [9, -2, 1, 7], [-1, 1, -23, -9], [21, 7, 11, -4], [2, 6, 7, -14], [-15, 11, 4, 16], [21, -11, 9, 9], [-1, -12, 0, -14], [-6, -14, 9, 1], [-4, 4, -16, -12], [3, 7, -18, 0], [11, 0, 19, 3], [-30, 0, -7, -8], [-4, 9, 17, -1], [3, -2, 2, 15], [-4, 11, -1, -6], [9, 14, -9, -1], [11, -2, -8, -13], [-15, -1, -14, 5], [-13, -10, 9, 3], [3, 7, 3, 14], [8, 10, -4, -6], [-21, 12, -4, -2], [17, -1, 7, 0], [6, -15, 13, 0], [11, -9, -9, 8], [-27, 11, -8, -1], [3, -9, -7, 2], [21, 7, 2, 2], [14, -4, -8, -4], [3, -7, -9, -8], [-6, -1, -3, -3], [15, 6, 7, 3], [14, -4, 16, 10], [-4, 13, -10, 3], [-7, 17, 2, 7], [-15, 16, 11, 7], [-18, -1, 3, 6], [-7, -14, 11, -9], [-9, 4, 6, -6], [11, 8, -4, -1], [-4, -12, 9, -10], [-30, 5, 3, -7], [-13, -11, -4, -9], [-6, 13, -13, 17], [-19, 2, -13, 7], [-13, 24, 8, -7], [-4, -8, -9, -10], [-12, 11, 5, 12], [-4, 0, -16, 4], [-9, -13, 14, 4], [6, -12, 10, -2], [-3, -18, 15, -16], [5, -4, 11, 3], [-7, 2, 4, 13], [-10, 10, -21, 0], [12, -14, -5, 13], [-1, 15, -12, -6], [21, 2, 0, 11], [14, -6, 2, -9], [-9, 11, -4, 11], [3, -2, -1, -1], [24, -9, -6, -3], [-28, 11, 0, -5], [-22, -8, -2, -4], [0, -22, 8, 2], [15, -5, -7, 4], [30, -10, 1, 16], [17, -7, 3, -1], [-16, -10, 6, 2], [-13, 6, -13, 17], [6, -4, 6, -6], [-18, 4, 8, -6], [29, -7, 9, -1], [15, 8, 2, -11], [-6, 9, 1, -2], [26, 15, -11, 5], [2, -13, -5, 17], [18, 9, 1, 18], [-10, -20, 3, -6], [-18, 18, -9, 10], [18, 12, 7, 0], [-4, -22, 9, -10], [-13, 14, -19, 1], [-27, -7, 4, -14], [12, -11, 14, 5], [-21, 16, -18, -17], [-4, -11, -2, 10], [35, -9, -4, 0], [0, -7, -8, 2], [62, -2, 1, 2], [-9, 21, -11, 6], [-28, 8, -13, 12], [-6, 17, -9, 11], [-33, 5, 6, 10], [-13, -7, -1, -10], [-3, 5, -5, 11], [14, -21, 11, 1], [32, 5, -7, 2], [-16, 3, 5, 12], [-3, -13, 2, -8], [-25, 10, 5, -7], [-9, -19, -9, 6], [2, -15, 0, 18], [12, -1, 14, -21], [21, 2, 16, 4], [-39, 7, 1, 9], [24, -10, 9, 10], [23, -13, 12, 4], [5, -10, -24, -5], [-1, 3, 18, 6], [-9, -12, 17, -5], [-21, 17, 12, 22], [38, -6, 9, 0], [-24, -4, -4, 8], [14, -10, 24, -11], [33, -9, 9, 10], [-46, -10, -12, 6], [-12, -9, 7, -4], [-9, -27, 16, -6], [-7, 2, -5, -13], [-33, -2, -13, -3], [9, -30, 3, 12], [-31, 21, -3, -5], [-7, 0, 0, 25], [-6, 5, 9, 13], [5, -8, 25, -3], [-16, 24, 2, 6], [42, -1, 7, -7], [11, 0, 5, -16], [-1, -12, -7, -19], [9, -3, 11, 10], [17, -6, -11, -12], [-18, 16, -7, -21], [-15, 32, 7, 0], [20, 3, 1, 19], [-9, 14, 1, 17], [-4, -12, -20, 6], [41, 5, -9, 4], [50, -3, 11, 3], [-19, 18, -11, -6], [11, 0, 16, -5], [5, -21, 9, 16], [3, 16, 1, 3], [26, 0, 11, -2], [11, -10, 0, -5], [6, -8, 3, 15], [0, -7, -3, -23], [-19, 19, 21, 9], [27, 3, 11, 0], [5, 22, -4, -3], [11, -17, 33, 8], [-27, 6, -6, -21], [-22, 30, 13, 8], [3, -10, -15, -15], [20, 4, -21, -13], [-3, -21, 23, -15], [-36, 20, -2, -17], [9, -23, 10, -22], [36, 12, 6, 2], [6, -12, 16, -17], [5, 3, -4, 2], [9, -1, 23, -3], [-1, -12, 22, 7], [-9, 21, 13, -4], [-27, 19, 14, 6], [11, 6, -11, 5], [-7, 8, 15, 15], [-12, 2, -23, -8], [-10, 13, 14, 12], [6, 5, -14, -15], [41, 10, -11, -2], [-7, 10, 14, 23], [-18, 2, -6, 18], [-34, 21, -1, -10], [51, 3, 20, -4], [26, -12, 35, 14], [3, -22, 19, 3], [-30, -7, -7, 2], [6, -13, -12, 4], [-7, 11, -3, -4], [-24, 10, 8, 8], [32, -10, -1, 3], [-15, -1, 0, -1], [0, -1, 7, 26], [-7, -6, 10, -1], [45, -15, 2, -2], [14, -20, -9, -8], [29, -15, 18, 6], [-9, 12, 17, -4], [8, 21, -16, 23], [-15, -4, 5, 13], [-28, 8, -2, 15], [-4, 15, -31, -9], [-7, -5, 7, 0], [15, -21, 14, 4], [-24, 1, -14, 13], [-4, 28, 14, -9], [3, 18, 17, 3], [-10, -13, -3, 8], [-18, -4, -17, 22], [-1, 10, -7, 7], [3, 6, 31, 15], [47, 1, 4, 16], [-24, 20, 1, 28], [-28, 9, -15, 4], [-28, 6, -8, -20], [14, -8, 7, -18], [36, 14, 6, -6], [9, -7, -2, 5], [23, -20, -3, 5], [30, 5, 10, -9], [48, -5, 7, -2], [26, -11, 6, -4], [59, 6, 9, 1], [18, 7, 21, -16], [0, -18, 3, 12], [38, 6, -2, 25], [-52, 9, -6, 17], [30, -21, 1, -28], [26, 20, 19, -6], [14, -6, 24, -9], [33, -22, 16, 19], [42, -7, -11, 3], [-21, -10, -18, 18], [-42, 0, -14, -4], [-9, 21, -15, -20], [26, 3, 3, -9], [-3, 0, -1, 4], [-30, -15, -4, 3], [21, 6, 1, 25], [41, -11, 18, 6], [17, 19, -8, 20], [30, 35, -15, 5], [38, -10, -10, 6], [8, 13, 25, -16], [45, -8, -14, -15], [5, -17, -1, -23], [12, -13, 27, -15], [12, -1, -18, 14], [-4, 1, 0, -21], [-13, 8, -9, -15], [9, -7, 32, 15], [-24, 2, -21, 13], [20, 15, -5, 25], [-12, 1, -10, -20], [-16, 6, -15, 4], [-21, -13, -10, 9], [-9, 21, 10, 12], [-22, -10, 2, 12], [-18, -15, -29, 11], [11, 11, 7, 21], [24, 5, 10, -5], [20, -8, 32, -17], [-13, 16, -21, 7], [-22, -27, 24, -7], [9, -23, 35, -10], [38, 0, 21, -26], [24, 10, -2, 10], [-48, 24, 6, -5], [23, -15, -18, 12], [-3, -13, 1, 11], [-19, -19, 0, -2], [-31, 12, -10, -6], [-6, 12, 0, -27], [8, 16, 11, 22], [30, -6, 24, 7], [41, 10, -13, -1], [-9, -7, 27, 8], [44, -7, -11, -19], [9, -6, 14, -17], [-21, -11, 15, -10], [5, -4, 25, 7], [5, -2, 4, -13], [-9, -34, 9, 0], [29, 11, 0, 0], [42, 8, -10, 6], [-25, -13, 0, 12], [-24, 26, 13, -12], [-60, 2, -1, 0], [8, 9, 4, 18], [5, -4, -16, -5], [-30, -29, -9, 2], [-3, 3, -13, -27], [8, 2, -3, 8], [21, -28, -20, 5], [12, 7, 29, -17], [-18, 20, -8, 19], [26, -15, -11, 1], [48, -8, -1, -6], [9, 3, 20, -23], [29, -34, 0, 1], [24, 5, 1, 27], [-12, 15, 3, 5], [-15, 18, -9, -10], [-25, 14, -21, 10], [-6, -27, -11, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1805_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1805_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_1805_2_a_o(:prec:=4)
    chi := MakeCharacter_1805_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_1805_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_1805_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6, 1, -6, -1, 1]>,<3,R![2, 5, -2, -3, 1]>],Snew);
    return Vf;
end function;