
// Make newform 8034.2.a.m in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_m();".
// 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_8034_2_a_m();".
// 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 := [-16, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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 := [[-1, 0], [-1, 0], [0, 0], [1, 0], [3, 0], [1, 0], [2, -1], [0, 0], [-3, 1], [0, 0], [4, 0], [-3, -1], [-1, 1], [-1, 1], [-5, 1], [2, -1], [0, -2], [-7, 1], [-8, -1], [-8, 2], [-6, 3], [0, -2], [0, 2], [-4, -2], [0, -4], [-5, 2], [1, 0], [4, 1], [-15, 1], [5, -1], [15, 0], [-12, 0], [9, -3], [-12, 1], [9, 0], [-12, 0], [8, 1], [3, 3], [2, -1], [21, 0], [-7, -2], [-2, -1], [4, -1], [-9, 3], [-4, 2], [-5, 0], [-3, -1], [-1, 4], [-12, 3], [4, -6], [13, 1], [8, -1], [5, 0], [-4, 4], [2, 2], [18, 1], [-18, -2], [-18, 0], [-20, 2], [6, 0], [15, -3], [10, -2], [-20, -1], [-3, 3], [-3, -3], [-18, -3], [6, -7], [-11, 2], [21, -1], [-2, 8], [10, -7], [-6, 3], [-10, 0], [9, 5], [6, 5], [3, -9], [-11, -4], [18, 4], [25, 1], [-4, 4], [9, -8], [-24, -3], [6, 2], [16, 0], [1, 0], [0, 8], [6, 5], [-1, -9], [19, -1], [-28, -2], [-25, 2], [-1, -5], [-6, -6], [-8, -5], [-18, 6], [11, -7], [9, 2], [-15, 3], [2, -9], [30, 1], [6, -6], [-2, 10], [-36, 2], [-9, -5], [-8, 0], [-3, 0], [0, 2], [18, -3], [6, 3], [-18, 6], [-42, 0], [-4, 3], [0, -2], [-4, 0], [-11, 8], [-32, 5], [-23, 3], [-15, 3], [-21, 3], [0, 5], [-6, -4], [-15, -2], [14, 2], [-24, -1], [8, 4], [-10, 2], [28, -6], [-18, -1], [-16, 3], [-33, -1], [1, -7], [8, 2], [-22, 6], [27, -3], [2, -8], [11, 6], [-11, -2], [-3, 3], [-34, 2], [-2, 10], [-6, -6], [-7, -10], [-19, -5], [5, -1], [12, -10], [26, 2], [-8, 9], [12, -3], [23, -9], [13, -5], [-27, 3], [-38, 2], [-2, -3], [-1, 11], [6, 3], [-2, -2], [-29, -6], [-19, -1], [4, 8], [6, -5], [7, -7], [2, -5], [12, -8], [0, -4], [17, 1], [-18, 3], [-12, 10], [-14, 0], [-32, -4], [19, 4], [6, 8], [-11, -7], [0, 4], [23, -6], [-12, 6], [16, -11], [16, -12], [4, -8], [-12, 0], [20, -3], [-48, 0], [3, -6], [5, -9], [8, -2], [-8, 14], [8, -2], [8, 1], [31, -1], [42, 1], [-4, 5], [6, 12], [12, -2], [8, -12], [-18, 12], [20, -10], [-8, -7], [48, 0], [-10, 1], [-16, 2], [6, 3], [-22, 2], [-38, 6], [28, -1], [-2, 11], [-8, -4], [44, 2], [-6, 16], [34, 4], [56, 2], [-42, 6], [19, 2], [45, 1], [-9, 9], [12, 4], [-42, -3], [10, 0], [26, -12], [15, -11], [-48, 0], [59, -1], [4, -3], [5, 3], [-28, 5], [-24, 12], [8, -14], [-38, -1], [-27, -7], [-19, 11], [-16, -3], [6, 0], [48, -6], [-18, 11], [-16, -3], [17, 5], [30, 7], [-9, 13], [9, 0], [30, -2], [-52, -2], [6, -15], [-2, 2], [12, 0], [-3, -14], [50, 0], [-14, 10], [-28, 14], [30, 0], [38, 8], [-9, -5], [27, -5], [40, 5], [15, -7], [-56, -4], [70, 0], [-2, -8], [35, 2], [13, 3], [60, 4], [14, -16], [2, 0], [-12, -6], [63, 0], [39, 3], [-2, 9], [1, 7], [-35, 9], [-30, -9], [28, -11], [34, -6], [-6, -11], [43, 3], [26, 7], [52, 6], [-34, 0], [39, 6], [45, 0], [-8, -2], [2, 3], [-18, -10], [-19, -1], [-39, -9], [-4, -6], [0, 5], [-46, 4], [-27, 7], [32, 10], [-2, -15], [15, -12], [8, -11], [42, 6], [18, 0], [-34, 4], [43, 5], [-4, 7], [19, -15], [7, 7], [4, 2], [40, 10], [40, 10], [12, 0], [-33, -2], [12, -4], [-36, 3], [-11, -1], [-15, 15], [-2, -10], [6, -8], [-35, -1], [-30, 3], [-30, -4], [-6, 6], [22, 5], [18, -6], [-12, 12], [-51, 3], [-39, 2], [-13, 14], [-16, 2], [78, -3], [-27, 15], [-23, -3], [69, -2], [-16, 4], [16, -9], [26, -14], [-24, -6], [-33, 9], [-56, 5], [-10, -7], [-42, 10], [36, 9], [58, -8], [-34, -6], [-20, 8], [-16, 12], [36, -2], [50, 4], [-22, -4], [14, -12], [57, 6], [42, 4], [10, 2], [36, 10], [-8, -1], [6, -6], [-32, -2], [12, 6], [5, 9], [29, -1], [-26, 10], [21, -21], [-70, 4], [-4, 8], [46, -4], [30, -16], [-39, 15], [-26, 2], [-84, -3], [-15, -15], [-17, -5], [30, 4], [16, -5], [68, -4], [41, -4], [-60, -4], [9, 0], [74, 2], [-12, -18], [-27, 9], [40, -5], [-12, -12], [0, -12], [-63, 1], [26, 6], [-10, 0], [-20, 10], [-47, 1], [-84, 0], [-28, 17], [-18, 0], [42, 13], [-1, 1], [-36, 3], [21, -7], [68, 0], [49, -11], [-28, 0], [-30, 12], [-34, -3], [-52, -10], [51, -6], [-22, -7], [-12, -9], [80, 3], [-14, -2], [58, 2], [-3, -13], [70, 2], [36, 6], [1, -11], [-14, -12], [-75, 5], [14, 0], [-6, 19], [-13, 2], [-40, 0], [42, -1], [-50, -5], [8, -8], [-22, 14], [6, 5], [0, 6], [-6, 10], [-13, 1], [-2, 15], [31, -2], [-12, -13], [-75, 6], [15, 9], [-5, 1], [-16, 10], [-4, 7], [-81, 5], [29, 10], [-45, -9], [-14, 2], [27, 0], [-66, 5], [46, 6], [42, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_m();".
// 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_8034_2_a_m(:prec:=2)
    chi := MakeCharacter_8034_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_8034_2_a_m();".
// 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_8034_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-1, 1]>],Snew);
    return Vf;
end function;