
// Make newform 819.2.a.k in Magma, downloaded from the LMFDB on 10 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_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_819_2_a_k();".
// 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_819_2_a_k();".
// 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 := [6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-3, 0, 1, 0], [-1, -4, 0, 1], [3, 2, -1, 0]];
    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_819_a();"
function MakeCharacter_819_a()
    N := 819;
    order := 1;
    char_gens := [92, 703, 379];
    v := [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_819_a_Hecke( : Kf := Kf)
    return MakeCharacter_819_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, 0], [0, 0, 0, 0], [1, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 1], [1, 0, 0, 0], [0, 2, 0, 0], [1, 1, -1, 1], [-1, 1, -1, -1], [-1, 1, -1, 1], [1, -3, -1, 1], [4, -2, 2, -2], [4, -1, -2, -1], [1, -1, -1, -1], [-1, -2, -1, 0], [-1, -1, -1, -1], [6, -3, 0, -1], [2, 4, 0, 0], [-6, 2, -2, -2], [-2, 3, -2, 3], [-1, -5, 3, -1], [1, 5, -1, 1], [-1, 2, -1, 0], [3, -2, 3, -2], [-5, 3, -1, -1], [10, -2, -2, 0], [8, -2, 0, -2], [-4, 0, 2, -2], [2, -6, 0, -2], [1, -1, 1, -1], [8, 2, 0, 2], [0, -2, 0, 2], [2, -5, 2, 1], [6, 0, -2, 0], [4, 1, 4, -1], [0, -2, 4, 2], [-4, 2, -6, 2], [-4, -2, 0, -2], [-15, 0, -3, 2], [4, 0, 4, 2], [-1, 7, -1, 1], [-2, 4, -4, 0], [-12, 2, -6, 4], [2, -4, 0, 4], [8, 7, -4, 1], [-4, 4, 0, 0], [1, -3, -1, -3], [3, 3, 1, 3], [2, -3, -4, -1], [-6, 0, 4, 0], [3, 5, 3, 1], [-6, -5, 6, -1], [3, -5, 7, -1], [2, -4, 2, -4], [14, 2, 2, 0], [-11, -1, 1, -3], [12, 0, 0, -2], [-8, -4, -4, 0], [11, 3, 3, -1], [12, -7, 0, -1], [8, -8, 0, -4], [5, -2, 5, -2], [-1, -3, -3, 1], [10, 0, -2, 0], [-10, 0, 0, 4], [2, -11, 2, -1], [-14, 2, 2, 2], [11, -5, 3, -5], [-4, -6, 2, 0], [3, 5, -5, 1], [16, 3, -2, -1], [6, -1, 6, -1], [2, 0, -6, -4], [-2, 0, -4, 4], [-2, -4, 2, 0], [-14, 1, 4, -1], [-6, 0, 0, 0], [3, 13, -5, 1], [-14, 1, -2, -5], [-1, 3, 3, -1], [2, -4, 2, 4], [12, -8, -2, -4], [4, -13, 4, -1], [4, 10, 2, -2], [14, 0, 6, -4], [7, -3, -5, 3], [-10, 5, -10, 3], [0, 0, -2, -4], [0, -3, 6, -3], [6, -6, -2, 6], [-6, 8, -6, 0], [-3, 10, -3, 0], [-10, 12, 6, 0], [-20, 0, -2, 2], [-24, 12, 4, 4], [-22, 0, 2, -4], [17, 0, -7, 0], [16, 4, 4, -2], [6, 4, -2, 4], [12, -12, -2, 0], [-1, 1, -3, 5], [-18, -1, -6, -3], [0, -8, 0, 4], [3, 5, 3, 1], [-3, -3, 7, 1], [2, 6, 0, 6], [-9, 2, 3, -4], [9, -4, 9, 0], [15, 1, 3, -5], [-10, 0, 0, 4], [2, 4, -6, 0], [0, 14, -2, -2], [12, -11, 0, -5], [-12, 0, 4, -4], [-2, 8, 2, 0], [-23, -3, 1, -3], [0, -4, 4, -4], [-18, -14, 6, -2], [-22, -2, -4, 2], [-7, -3, -7, -5], [15, -3, 7, -7], [11, -5, -9, -1], [22, 4, -2, 2], [-2, -5, -2, 7], [19, -3, 5, -3], [5, 1, 5, 5], [-2, 2, -4, 6], [2, 16, 2, -4], [-6, 10, -2, 2], [-17, 3, -1, -1], [-34, 2, -6, -2], [-22, 3, 2, 3], [17, -9, 3, 7], [-29, 3, -1, 3], [-23, 2, 1, -2], [-11, -9, 5, 3], [8, -13, 2, -1], [-3, 7, -5, -1], [-2, 8, -2, 6], [13, -1, 13, -1], [-16, 2, 12, 2], [0, -7, 0, -1], [-8, 12, 8, 4], [-14, 7, -2, -1], [2, 4, -12, -4], [-2, 17, 4, -5], [5, -1, 9, 3], [14, 10, 2, 0], [2, -6, 6, 6], [-18, 9, 6, 1], [12, 10, 10, -6], [-6, 2, -6, -4], [0, -10, 16, 2], [-8, 4, 4, 0], [-3, 9, -5, 9], [-15, 9, -3, -1], [-4, 10, 0, -2], [5, 2, 5, 6], [-16, 10, 6, 6], [-13, 8, 11, 4], [16, -13, 4, -1], [-19, 3, 5, -5], [18, 4, -2, -4], [2, -2, 2, 6], [20, -1, -4, 1], [17, -2, 5, 0], [-4, -20, 0, 0], [32, -6, 6, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_819_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_a_k();".
// 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_819_2_a_k(:prec:=4)
    chi := MakeCharacter_819_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(997) - 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_819_2_a_k();".
// 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_819_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6, -5, -7, 1, 1]>,<5,R![24, 20, -10, -3, 1]>],Snew);
    return Vf;
end function;