
// Make newform 2034.2.a.s in Magma, downloaded from the LMFDB on 29 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2034_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_2034_2_a_s();".
// 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_2034_2_a_s();".
// 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, -1, -12, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [5, -10, -2, 1], [-1, -12, -1, 1]];
    Rf_basisdens := [1, 1, 2, 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_2034_a();"
function MakeCharacter_2034_a()
    N := 2034;
    order := 1;
    char_gens := [227, 1585];
    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_2034_a_Hecke( : Kf := Kf)
    return MakeCharacter_2034_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 := [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 0], [1, 0, 0, -1], [-1, 1, 0, 0], [2, 0, 0, 0], [1, -2, 0, -1], [1, 1, 0, 0], [-1, 1, 1, 0], [0, 0, 1, 0], [3, 1, -2, 2], [3, 1, -1, 2], [4, -2, 0, 2], [3, -2, 0, -1], [-1, 2, 1, 1], [-3, 0, 0, -3], [0, -2, 4, -2], [0, 2, 2, 0], [3, -2, 0, -1], [-3, 2, -1, -1], [6, -2, 0, 2], [-2, 2, 1, -2], [1, -2, 0, -1], [-1, 3, -2, 2], [-1, 2, 2, -1], [4, 0, 1, 4], [0, 2, -1, 0], [4, -2, 0, 2], [2, 2, -2, 2], [1, 0, 0, 0], [-7, -1, 0, 2], [7, 0, 2, 1], [2, 4, -2, 0], [-4, -2, 2, 4], [-1, 5, 0, 4], [-6, -4, -1, 0], [-4, 2, 0, -4], [0, -6, 0, -2], [2, 0, 1, -4], [-1, -3, 4, -4], [6, -2, -2, 4], [1, -2, -1, -3], [3, 5, -1, -4], [0, 2, 2, 0], [4, -4, 3, 0], [-6, -2, 3, 2], [-10, -4, 0, 2], [2, -2, 1, -2], [7, -4, 6, -3], [4, 4, -3, 0], [-10, -2, -4, 0], [8, -2, 0, 6], [-5, 5, 2, 4], [-7, 3, -6, 2], [-6, 0, 4, 6], [-6, 0, -5, 0], [-14, 4, -1, 2], [-8, 2, -3, 4], [2, -6, 6, -6], [9, 1, 2, -2], [10, 0, -2, -4], [-6, 0, 5, 6], [-4, 2, -8, 2], [6, -2, 2, 4], [3, 1, -2, 8], [-21, -3, -2, 0], [-12, 2, 2, 0], [-1, -2, 2, -5], [8, 4, 4, 0], [5, -1, 3, 2], [4, -2, 0, 2], [-25, 2, 1, 1], [3, -7, 2, 6], [24, -4, 1, 0], [-15, -3, 6, 4], [-8, -2, -4, 2], [2, -8, 0, 0], [13, 2, 3, 1], [4, -6, 4, -8], [6, -6, 6, -2], [-12, 2, -2, 2], [4, 4, 2, 6], [5, -2, 3, 3], [-2, -4, 2, -8], [-15, 4, 0, -1], [-9, 2, 0, 5], [-7, -3, 8, 2], [-10, 0, 6, 0], [-1, 8, 0, -5], [-3, 8, -4, 3], [-6, 2, -2, 2], [3, 1, 3, -8], [-2, 6, -3, 2], [22, 2, -2, 0], [13, 2, -6, 1], [-16, -2, -2, -6], [-21, 4, 2, -5], [26, -2, 0, 0], [-3, -3, -2, -8], [1, -2, -3, -3], [-18, 0, -2, 4], [3, 1, -8, 4], [-5, 5, -2, -6], [-2, -10, 4, 0], [3, -1, 2, 12], [10, 2, 0, -2], [-1, 4, -2, -3], [6, -6, -4, 0], [-2, -4, -9, 0], [1, -1, 2, 10], [-4, 2, -3, 8], [9, -2, 3, -7], [14, 8, 4, -2], [7, -10, 0, -5], [-16, -2, 7, 4], [19, 2, -4, 9], [3, 7, 0, -4], [-10, -2, -2, 0], [-5, -4, -4, 3], [10, 4, -2, 2], [-3, 7, 3, 2], [6, -6, 6, -2], [6, 4, -4, -8], [10, -4, 2, 6], [6, -8, -4, -10], [2, 6, -3, 8], [6, 6, 4, -2], [-10, 4, -4, -6], [16, 4, -2, 12], [16, 4, 5, 0], [14, 2, 2, 2], [-11, 11, -5, 0], [22, 0, -7, -4], [9, 3, 1, -2], [14, 0, -4, -4], [37, -1, 2, -2], [9, 3, 0, 0], [12, -12, 10, -2], [22, -4, -3, 6], [6, 6, -4, 0], [5, 6, 4, -3], [19, -12, 2, -5], [5, -1, -10, 2], [-7, -6, 0, -13], [3, 2, -7, 3], [-6, 0, -9, 0], [-12, -8, 0, -10], [15, 6, 4, -3], [7, 10, 2, 3], [-24, -8, -2, -8], [11, -6, 7, -9], [23, -9, 2, 2], [15, 11, 0, 12], [13, 7, -7, 8], [8, -8, 0, -8], [18, 6, -4, 12], [-19, 0, 6, 3], [-20, -6, -8, 4], [-4, -2, 4, -8], [26, 0, -3, 2], [-4, 0, -14, 8], [4, -4, 0, 6], [12, -4, 2, -12], [-6, 2, 0, -4], [2, -8, 6, -12], [-11, -2, -1, 11], [-13, -4, 4, -7], [-9, 10, -9, 11], [-23, -2, 10, -3], [5, -1, -6, -8], [1, -9, 2, -2], [38, 0, 0, 0], [-8, 12, -8, 4], [9, -1, -6, 6], [-12, -8, 2, -16], [16, 12, 4, -2], [0, 0, 10, -8], [18, 4, -8, 16], [0, -8, -1, -10], [-12, -2, -6, 8], [-10, 8, -1, 8], [26, 2, 4, 4], [0, 4, 9, 8], [-17, -5, 6, 2], [11, -7, -5, -8], [-2, -4, 9, 6], [-12, 4, -6, 2], [20, -4, -2, 2], [-13, 0, -2, 3], [-13, -3, 5, 8], [4, 8, -12, 10], [-2, -4, 6, -6], [4, -4, 8, 4], [7, 1, -8, 8], [-27, 3, 2, 6], [22, 4, 8, -4], [30, -2, -8, 2], [-8, -2, -8, 0], [-6, 6, -16, 6], [-17, 13, -3, -4], [11, 12, 0, 11], [10, 10, -9, 6], [9, 10, -3, -7], [-18, 0, 4, -8], [1, -4, 14, 3], [18, -16, 5, -10], [-5, -4, 10, -11], [-4, 4, 8, 12], [26, -8, 0, 0], [9, -3, -6, 4], [-14, 14, 2, -2], [-8, 4, 5, -4], [12, 4, 6, -4], [-14, 14, -10, 6], [-11, 7, 1, -16], [18, -12, 10, -8], [-30, 4, -3, -4], [15, 11, 4, -10], [-20, 0, -14, 4], [-8, -2, -5, 14], [30, 0, -8, 4], [-6, 4, -9, 8], [12, 8, -12, 14], [8, -12, 11, -6], [10, -16, 2, 6], [15, 1, 9, -10], [12, -2, -4, -8], [-8, 8, -6, -12], [5, 15, 0, -6], [13, 12, -6, 7], [2, 6, 4, -6], [-13, 2, -10, -1], [7, -3, 8, 2], [15, -9, -6, -6], [-8, -2, 2, 0], [-10, -4, 14, -14], [-6, -10, -6, -6], [-16, 12, -5, -4], [-9, 6, -20, 9], [41, 3, 1, 8], [-8, 2, 2, -6], [-16, 0, -6, 6], [-26, 10, -5, -6], [32, 0, -9, 12], [25, 5, 2, -6], [-7, -5, -1, -12], [-43, -4, 4, -1], [17, -4, 14, -5], [20, -12, 2, 12], [-14, 6, -2, -8], [-22, -12, 6, 0], [5, -2, 2, 3], [-42, 0, 4, 6], [-2, 18, -4, -10], [-23, 0, 4, 1], [-20, 12, -4, 4], [5, -10, 7, -7], [9, -22, -2, -15], [10, 0, -5, -2], [0, -18, 6, 4], [-45, -3, -6, -2], [19, -7, 4, 0], [-10, 6, 0, -6], [-8, 12, 4, -10], [16, 12, 4, 16], [18, 4, -2, 8], [-15, -13, -2, 8], [-30, 4, 4, -2], [8, 12, -2, 6], [-16, 12, -11, 8], [11, -11, -5, -2], [-18, -6, 4, -8], [-20, 10, 0, -12], [13, -5, -8, 6], [19, -3, 6, -10], [-13, -3, 12, 6], [-20, 2, 4, -6], [-21, 1, 1, 2], [31, 5, -2, -2], [12, -16, -2, -16], [4, -16, 14, -12], [10, -10, -5, 10], [30, 8, -5, 20], [18, 14, -10, 6], [-9, -5, 6, 8], [18, 2, -6, -4], [-8, 14, 6, -8], [-1, 4, -4, 15], [0, 12, -11, -2], [8, 12, -16, 8], [-8, -6, -13, 10], [4, -4, 4, -12], [-38, -6, 0, -14], [-38, -6, 12, -20], [33, 7, -15, 14], [-25, 25, -12, 12], [-9, 4, -4, -7], [17, -5, 8, -12], [-20, -14, -6, -10], [-32, 0, -8, 8], [19, -3, -10, 8], [28, 2, -13, 6], [14, 0, 7, 0], [-39, -9, -4, 0], [-29, -10, 4, -17], [4, -8, -6, -18], [13, -13, 8, -12], [-7, 11, 7, -10], [-15, -2, 1, -5], [-1, -18, 13, -9], [-19, -18, -5, 11], [-28, -10, 5, -2], [13, -3, -12, -14], [14, -18, 0, -12], [-18, -8, 6, -2], [4, -2, -12, -12], [10, -8, -10, -10], [-4, 14, -12, -2], [18, -18, 4, -14], [10, -6, 16, -2], [6, 22, 6, 2], [22, 2, -10, -14], [-30, -10, 13, -16], [-27, -17, -14, 2], [-19, -11, 12, -18], [31, 8, -18, 19], [38, 8, -2, -4], [-16, 4, 8, -8], [-30, 4, 8, 10], [30, -4, 7, 2], [21, -22, 11, -3], [-2, 20, -13, 6], [24, -16, 9, -6], [-20, 6, -14, -2], [30, 4, 0, 2], [37, 7, 6, 2], [6, -22, -4, -6], [50, 0, -8, -4], [2, 18, 4, 6], [0, 2, 5, 12], [-7, 6, -9, 9], [1, -10, 0, 3], [-35, -4, 12, -15], [5, 4, 14, 7], [-26, -4, -1, -6], [-12, -18, 0, -6], [-34, -10, 10, 2], [8, 8, -10, 14], [-8, 10, -4, -10], [1, -8, 4, 5], [10, 2, -10, 10], [-35, -5, 6, -18], [13, 8, -8, -3], [-26, -10, 3, -2], [14, 6, 6, 6], [11, -13, 14, -2], [36, -6, 6, 6], [40, -2, -4, -10], [10, -4, 8, -18], [-46, 10, -2, 0], [-4, 8, -4, -22], [43, 3, 2, 10], [1, 22, 7, -1], [44, 0, 4, -4], [7, 15, -10, 20], [-21, -14, 2, -25], [13, 11, -12, 12], [66, 0, -1, 4], [20, 18, 2, 0], [26, -6, 4, 8], [-34, -10, 2, -10], [-6, -14, -2, -8], [7, -12, 2, -5], [20, 2, 1, 8], [43, 2, -15, -5], [-18, -18, -6, 0], [10, 8, -2, 18], [18, 16, 0, 8], [-37, 2, -4, 13], [21, 6, 10, 1], [47, 5, 4, -8], [-23, 26, -4, 15], [-56, 8, -3, 10], [-31, -3, 10, -22], [22, 0, -4, 14], [30, -22, 8, -4], [33, 11, 2, -6], [7, 6, -5, -13], [-5, 15, 8, 8], [14, 8, -8, -14], [-48, 0, -8, 10], [42, -14, 9, 4], [25, -24, 2, 7], [5, -2, -8, 1], [20, 0, 10, 8], [-40, 10, -4, -14], [21, 7, -6, 2], [-31, -10, -4, -5], [-27, 10, -10, 5], [-36, 18, -9, -6], [-4, -4, -1, -4], [-28, 0, 16, 2], [-20, 0, -17, 2], [-11, 14, -8, -9], [66, 4, 4, 4], [44, -4, 9, 2], [34, 8, 10, 4], [5, 2, 24, -17], [50, 8, -2, -4], [18, -8, -8, -14], [-46, 16, 0, 4], [-6, -6, 15, -12], [-61, 5, 2, -14], [-6, -10, -4, 0], [-7, -4, 14, -19], [-7, 8, -12, 1], [22, -16, -7, 24], [-4, 4, -18, 16], [10, 10, -20, 2], [-49, -10, 11, -23], [-6, 22, -9, 4], [-13, 13, 1, 22], [20, 22, 8, 18], [51, 9, -10, 6], [23, 8, 10, 11], [44, -10, -5, 22], [-7, -6, -12, -13], [-44, -16, -8, 12], [27, 3, -14, 28], [-4, -6, 10, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2034_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2034_2_a_s();".
// 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_2034_2_a_s(:prec:=4)
    chi := MakeCharacter_2034_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_2034_2_a_s();".
// 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_2034_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![24, 0, -18, 0, 1]>,<7,R![-8, 16, -3, -5, 1]>,<11,R![-12, -24, -9, 3, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;