
// Make newform 1805.2.a.g in Magma, downloaded from the LMFDB on 03 March 2026.
// 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_g();".
// 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_g();".
// 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 := [7, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    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_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 := 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]];
    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, 1, 1], [-1, 0, 0], [0, 1, 1], [-1, -1, -1], [5, 0, 0], [0, -1, 2], [0, 0, 0], [1, -1, 2], [0, 1, 1], [0, 1, 0], [-2, 3, 5], [2, -3, -1], [0, -2, 1], [2, 1, -1], [-6, 3, 4], [-2, -1, 1], [-3, 2, -2], [6, 0, 2], [8, 2, 3], [-7, 1, -2], [8, 2, -2], [-2, 3, 0], [6, 0, -4], [-3, -1, 3], [-4, 5, 4], [-1, -1, -6], [-5, 1, 4], [4, -6, -5], [1, -1, -7], [-2, -6, 0], [6, -5, -5], [-6, 0, 5], [-1, 0, 2], [-3, 1, 2], [11, 4, 0], [5, 0, -1], [8, -1, -6], [-10, 1, 1], [2, -3, -3], [15, -1, -2], [-1, -6, 2], [-8, 6, 0], [10, -1, 2], [6, 5, -1], [4, 0, -3], [12, -4, -4], [-5, 9, 3], [-4, 0, 0], [7, -5, -7], [3, 10, 1], [8, 1, 8], [10, 4, 2], [-1, 1, -7], [-7, -3, -1], [4, 6, 2], [-13, 8, 2], [-3, -1, 0], [-2, 10, 8], [-3, -4, -1], [5, -2, 3], [-5, -1, -4], [-7, -5, -5], [0, -5, 1], [-23, -5, 2], [6, -2, -3], [-5, 5, 3], [20, -8, -6], [0, 4, -3], [1, -6, -5], [-22, 3, 2], [-14, -3, 8], [19, -12, -10], [4, 0, -7], [12, 4, -1], [-16, 12, 7], [-15, 11, 3], [8, -7, -7], [-14, 4, -3], [-9, 9, 0], [-24, 1, 0], [15, -3, -1], [8, 5, 7], [-7, 3, -5], [1, -5, -17], [8, -13, -9], [-9, -3, -8], [-6, 15, 4], [6, 1, -5], [15, -5, -8], [-1, -6, -2], [-37, 0, 1], [26, -1, 0], [-25, -3, -6], [8, -6, -3], [17, -1, -1], [4, -13, 0], [-1, 10, 13], [9, -17, -7], [0, -4, -3], [30, -3, -4], [14, 2, -3], [3, -16, -10], [6, -6, 6], [7, -10, 0], [-6, -9, 6], [7, 10, 9], [12, -4, -14], [-7, 10, 0], [0, -4, -6], [-9, 1, 14], [11, -9, -10], [-17, -7, -8], [7, 9, 1], [16, -8, -14], [4, -11, -16], [-12, -7, -5], [2, -16, -11], [19, -5, -3], [13, -5, -12], [-14, 15, 3], [-1, 7, -3], [-8, -2, 3], [25, -3, 7], [-16, 19, 7], [27, -7, -7], [2, -17, -11], [1, -14, -14], [-6, -2, -5], [-29, 5, 11], [6, 19, -2], [-8, -14, -1], [14, -5, -7], [23, 8, -4], [0, -6, 9], [0, 11, 12], [-9, -12, -7], [20, -1, 0], [-29, 1, 1], [12, -11, -6], [11, 2, 6], [-1, 8, 11], [28, -4, -12], [9, 7, -3], [-2, -6, -15], [-6, -3, 2], [-27, 5, 2], [12, 11, 5], [4, 4, 15], [-14, 16, 1], [20, 1, 13], [4, 6, -10], [-20, -10, 7], [11, 10, 3], [18, 4, -4], [11, -6, -6], [-1, -6, 17], [-4, -14, -13], [-1, 12, -1], [-2, 6, -8], [21, 4, 1], [-7, -11, -17], [-46, -4, -4], [-17, 4, 5], [-5, 3, -6], [12, -8, 1], [-12, 2, 6], [-3, 4, 10], [-34, 6, 2], [-2, 14, 13], [-26, 16, 17], [-4, 4, 3], [-15, 0, -2], [-28, 17, 9], [6, -9, -16], [21, -21, -22], [-16, -12, 6], [25, -13, -2], [-2, -16, -11], [0, 11, -5], [-10, 15, 10], [-2, 16, 23], [-24, 8, -4], [-15, -9, -3], [-4, -8, -6], [-19, -1, 8], [-18, 14, 11], [16, -5, -17], [-10, -23, 4], [1, 6, -17], [15, -5, -21], [-18, -1, -17], [35, 3, 13], [36, -6, -8], [20, 8, -5], [-11, 12, 14], [1, -10, 4], [-6, -10, -16], [-26, 0, 8], [7, 14, 11], [-22, 19, 20], [22, -21, -16], [10, -2, -8], [-28, 14, 21], [44, -20, -14], [16, 6, 26], [2, 7, 8], [-14, 19, 1], [14, -4, 5], [-1, 1, -4], [1, -5, 2], [34, 4, 1], [0, -9, 10], [28, 3, 8], [2, 15, 22], [-14, -15, 2], [-2, -4, 5], [-25, 23, 17], [-11, 6, -16], [44, -13, -11], [12, 13, 5], [11, 27, -3], [-8, -9, 5], [-28, 7, 16], [3, 6, -15], [0, 1, 2], [-31, -11, 6], [25, 1, 9], [-25, 4, 1], [-26, 5, 0], [-35, -9, -2], [16, -12, -5], [22, 4, 11], [-7, -11, -35], [34, -24, -19], [-31, 8, 7], [-2, 9, -1], [-3, 18, -4], [9, 6, 14], [12, -7, 0], [-14, 17, 24], [-19, -11, -1], [2, -13, -13], [-19, -13, 3], [-20, -4, 11], [-1, 5, 11], [22, -7, 10], [36, -28, -12], [-15, -17, 11], [-41, 20, 17], [8, -8, 12], [-6, 14, 21], [5, 11, -1], [-24, 4, 23], [-22, 6, 5], [-23, 11, 8], [42, -13, -8], [8, -3, -23], [-41, 13, 12], [28, -18, -13], [18, 8, 34], [-13, -3, 6], [19, -21, -22], [-2, -18, -15], [11, -20, -5], [-41, 15, 12], [-33, 32, 29], [5, 21, 6], [-29, 13, 8], [6, 4, -11], [-30, 2, -2], [31, -2, -13], [15, -25, -2], [-25, 7, 14], [16, -5, 19], [-19, 22, 23], [48, -16, -19], [60, 5, 8], [2, -15, -7], [49, 6, -5], [-43, 17, 20], [16, -13, -28], [-9, 0, -9], [-30, 27, 15], [11, -6, 3], [-36, 17, 7], [-18, -7, -22], [-14, -4, -17], [7, 2, 3], [24, 11, 4], [-1, 11, -11], [35, 9, -13], [-34, -12, -10], [15, -21, -17], [1, -9, -17], [20, -3, 0], [-14, 13, -5], [51, 8, -7], [-5, 3, -1], [-42, 0, -7], [18, -2, 9], [32, -1, 2], [-20, 21, 5], [-1, -2, 8], [-25, 3, -6], [50, -21, -7], [-16, -2, 16], [22, 10, 27], [-15, 25, 19], [52, 7, 14], [0, -16, -41], [-10, -16, -3], [24, 27, -3], [27, -22, -31], [-48, 14, 22], [13, 12, -5], [6, -6, 28], [4, 0, -31], [38, -7, 11], [-23, -5, 15], [10, -5, 22], [-29, -7, -29], [21, -7, -6], [-9, -2, -22], [-32, -10, -9], [9, 1, -25], [51, -17, -10], [55, -4, -7], [-11, 7, -11], [-9, 2, -8], [3, -5, 17], [24, -10, 13], [24, 24, -7], [-16, -17, 10], [34, 2, -7], [-8, 16, 19], [-9, -3, 27], [8, 25, -8], [-2, -2, 3], [-44, 34, 26], [16, -7, -14], [6, -26, -4], [8, -12, -20], [28, -30, -9], [1, -16, -2], [12, -28, -19], [24, 9, -11], [14, 0, -17], [60, -8, 3], [-28, -9, -34], [-3, -23, -1], [17, 14, 0], [16, -24, -31], [-32, 1, -2], [-45, -7, -7], [2, -11, -19], [-21, 3, -24], [1, 10, -16], [30, -22, -36], [5, -8, -17], [-25, -7, 16], [33, 13, 22], [9, -26, -18], [-30, 4, 17], [27, 5, 7], [-9, -11, 5], [37, 9, 18], [-28, -2, 20], [30, 30, -7], [10, 1, -17], [-8, -9, -4], [35, -17, -6], [8, 15, -9], [-3, 0, 30], [22, -12, -5], [6, 8, 22], [5, 6, 35], [19, 11, -14], [34, -7, -11], [4, 15, 6], [6, 18, 18], [-9, -5, 1], [-46, 8, 14], [51, -21, -22], [46, -25, -4], [11, 8, 14], [31, 13, 29], [48, 10, -3], [-19, 8, -12], [-39, 24, 13], [64, -1, -12], [38, 14, -12], [-21, -1, -11], [-41, -1, 13], [-6, -13, -22], [1, -7, -11], [33, 4, -2], [1, -19, -7], [34, 5, 24], [29, -17, -13], [-44, -16, 3], [-41, 22, 25], [69, 1, -8], [18, -25, -35], [9, -6, -17], [-8, 12, 3], [-38, 31, 11], [-11, 3, 0], [31, 21, -11], [-32, 24, 38], [-47, 25, 25], [32, 13, 14], [8, 17, 6], [28, 5, -1], [18, 5, -23], [-18, -16, -1], [-52, 23, 16], [-48, -5, 2], [32, 7, 15], [-61, 1, -8], [7, -5, -9], [19, 9, -12], [6, -13, 14], [84, -12, -10], [12, -21, -1], [11, -13, 2], [-19, -35, 6], [39, 9, 2], [9, -11, 14], [42, 16, 17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1805_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1805_2_a_g();".
// 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_g(:prec:=3)
    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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-7, -6, 1, 1]>,<3,R![-7, -6, 1, 1]>],Snew);
    return Vf;
end function;