
// Make newform 4235.2.a.u in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4235_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_4235_2_a_u();".
// 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_4235_2_a_u();".
// 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, -2, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -6, 0, 1], [-2, 5, 1, -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_4235_a();"
function MakeCharacter_4235_a()
    N := 4235;
    order := 1;
    char_gens := [2542, 1816, 2906];
    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_4235_a_Hecke(Kf)
    return MakeCharacter_4235_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, -1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [-5, 0, 1, -1], [0, 0, 1, 2], [-1, 0, -2, -1], [-1, 1, 1, 0], [-4, -2, 2, 0], [-2, -1, 2, 1], [-3, 5, 3, 0], [-4, 3, 0, 1], [-3, 3, 3, 0], [1, 4, -1, -1], [9, 1, -1, 0], [-4, 1, 0, -3], [1, 3, 0, -2], [5, -3, -3, 0], [2, -2, -2, 2], [0, -4, -3, -2], [2, -3, -1, -1], [-1, 2, 6, -1], [1, -4, 2, -1], [-7, 0, -2, 1], [-6, 3, -2, -1], [-6, -4, -3, 2], [6, -4, -4, -2], [-4, 0, 2, 4], [-4, 4, 0, 0], [-4, -4, 0, 2], [-8, -4, -4, 0], [5, -7, -5, -4], [0, 4, 2, 0], [-6, 2, 4, 2], [4, 3, 3, 5], [1, 0, -2, 1], [-7, -1, -1, -6], [1, 8, 2, -1], [2, 2, 3, 2], [-4, 0, 4, 0], [-9, 0, -2, 1], [-2, 8, 2, 0], [-5, -7, 1, -4], [3, 3, -5, 2], [-3, -5, 0, 0], [2, -4, -6, 0], [-6, -8, -1, 2], [-10, -4, 4, 6], [6, -2, 0, 6], [3, 7, 5, 2], [6, 2, 0, 2], [0, -1, -10, 1], [-8, -3, 0, -3], [5, 6, -4, 3], [-10, -2, 2, -2], [-17, -4, 0, -7], [-2, -4, -6, -6], [7, -9, -5, 4], [-12, -2, -6, -4], [-6, -2, 0, -4], [0, 6, -3, 4], [1, 2, -4, 5], [-3, -3, 4, -2], [6, -6, -8, -2], [4, -8, -4, -6], [8, 8, 2, 6], [1, -1, 1, -2], [11, 7, 3, -2], [-7, 7, -2, 2], [14, -2, -6, 2], [-4, 1, -1, 5], [11, 6, 7, 7], [-21, 1, 1, 0], [0, 2, 6, 0], [-8, -6, 5, 2], [-2, -4, 10, 2], [-11, -2, 6, -1], [0, 8, 12, 2], [-13, 3, -4, 0], [-11, 5, 2, 6], [0, -5, 5, 7], [0, -9, -3, -5], [10, 8, 4, 8], [-14, 6, 4, 0], [9, -3, 7, 0], [-8, -11, 7, -3], [7, -5, -9, -6], [-6, 5, -4, 1], [7, -17, -1, -4], [10, -8, -9, 6], [5, 10, 8, 7], [-5, -1, 7, -2], [10, -11, -5, -1], [-4, -10, -2, -4], [29, -2, -4, 1], [6, -6, -6, -6], [-3, -6, 2, 1], [11, -6, 10, -1], [-2, 4, -6, 0], [10, 8, -4, -4], [-23, 7, 7, -2], [-8, -6, 0, 2], [-6, 0, -14, -2], [10, -2, -4, 2], [13, -2, 14, 3], [-2, -10, -5, 0], [11, 4, -3, -5], [-12, 14, 6, 0], [-4, -13, 6, -3], [-1, 8, 10, 1], [-11, -1, -15, -4], [13, 7, 13, 10], [-23, -1, 0, -8], [18, 2, -2, -4], [16, 9, -13, 5], [-25, 6, 5, -1], [13, -6, -3, 9], [10, -8, -8, -2], [6, -9, -13, -9], [-7, 12, 4, -1], [-7, 15, 5, 6], [5, -14, -15, -1], [-5, -5, -11, 2], [-27, 5, 4, 6], [0, 2, -8, -2], [0, -8, -4, 2], [-7, 7, 2, 0], [19, 10, -3, -5], [-10, -12, 3, -4], [-8, 0, 8, 4], [16, 2, 2, -6], [2, 2, 4, -8], [12, -4, -8, -12], [-17, -13, -6, 4], [22, -7, -6, -3], [-21, -4, 2, 7], [10, 10, -2, 0], [-3, 0, -6, -11], [-16, 8, 12, 0], [3, 10, -6, 1], [8, 10, 12, 8], [-11, 21, 15, 10], [11, 11, 15, 6], [-4, 0, -4, -2], [1, 1, -10, 6], [-23, 8, -1, 1], [4, -10, -11, -16], [4, 9, 14, 5], [-5, 5, 3, -6], [-1, 11, 9, 6], [-3, 2, -6, -3], [7, -5, -3, -2], [2, 4, -18, -6], [5, -13, -17, 0], [2, -10, 6, 2], [10, -6, 0, 2], [19, 12, -2, 1], [-15, 4, 7, 5], [-21, 1, -8, -6], [-35, -9, 5, -6], [1, 1, 5, -4], [-25, 17, 9, 4], [7, 1, -6, 4], [4, -16, 4, -4], [-5, -16, -5, -11], [6, 0, -6, 6], [2, -16, -13, -8], [12, 18, -8, 4], [11, -8, 2, 3], [2, -2, -20, -2], [-19, -4, 0, -1], [-10, 11, 3, 7], [5, -1, -11, 0], [1, -11, -18, -18], [4, 6, -2, 0], [24, 3, 13, -1], [-12, 1, 7, -7], [1, -14, -10, -11], [-26, -13, 4, 7], [15, -3, 15, 6], [5, 0, 16, 9], [20, -8, 0, 8], [33, 2, 7, 3], [1, 11, -3, 0], [-1, -12, -18, -7], [-37, 11, 7, 8], [1, -10, 11, 1], [-16, 0, -8, -6], [21, 4, -12, 9], [6, 22, 6, -6], [-20, 2, 6, 2], [-8, -2, 4, -2], [-11, 8, 6, -1], [-3, 3, -1, 0], [-9, -6, 8, 1], [-24, 10, 10, 4], [-9, -9, -7, -8], [-1, -4, -21, -13], [32, 4, 0, 16], [-12, -6, -12, -4], [-35, 4, 6, -11], [16, 2, 12, 12], [-33, 9, -2, 2], [-6, 1, 10, 7], [-10, 0, -6, -12], [16, 13, 14, 1], [-42, 14, 2, 2], [-16, -10, -8, -8], [-2, 9, 20, 3], [7, -17, -7, -2], [-22, 4, 12, -8], [11, -3, 3, 2], [5, -16, 1, -5], [9, -12, 4, 1], [-13, 26, -4, 3], [5, -3, -11, 10], [-24, -13, 2, -7], [-7, 1, 9, 0], [8, 16, 16, 0], [-12, -12, 6, 12], [-5, 22, 16, 13], [-8, 17, 3, -11], [1, 3, -11, -12], [5, 4, -2, -9], [-28, 14, 12, -6], [-4, -6, -2, -8], [16, 6, -14, -14], [12, 16, 2, 12], [4, -19, -23, -5], [-33, 5, 7, -2], [16, -14, -12, 10], [-26, -4, 4, -2], [-18, 16, 8, 10], [-9, 20, -9, 1], [-11, -20, 10, 7], [0, -16, 6, 2], [23, -1, -9, 10], [0, 16, 0, 2], [43, 1, 4, 0], [39, -15, -15, 10], [-38, -4, 12, -6], [-25, 14, 7, 3], [-8, 27, 19, 1], [7, -6, -9, -11], [-9, -6, -10, -15], [-47, 8, 11, -9], [-5, -7, 10, 4], [24, 14, 10, 2], [-6, 0, 16, 4], [-1, -13, -9, -18], [-23, 23, 24, -6], [-17, 5, 13, 0], [17, -24, -16, 3], [44, -18, -9, -4], [20, 12, -4, -12], [-6, 3, -13, -1], [5, 0, -10, 7], [12, 14, -2, 12], [-35, -10, 7, 3], [-35, -11, 9, 6], [28, -26, -10, -2], [-20, -17, 0, 9], [-4, 22, -11, 4], [-7, 6, 4, 11], [-5, -3, 12, 0], [6, -5, 7, -5], [-10, -14, -26, 2], [11, -5, -1, 2], [-21, 15, -11, -4], [22, 16, -6, -4], [11, -11, 17, -2], [10, -12, -20, -8], [-16, 18, -2, -2], [16, 4, 3, -2], [22, -6, -17, 8], [37, 13, -5, 6], [-30, -6, 10, -12], [18, 2, -8, -10], [8, -6, -4, -6], [-17, 16, 12, 9], [42, 4, 18, 2], [9, 0, -12, -5], [-45, -13, 4, -12], [15, 4, 2, 9], [8, 14, 6, 0], [29, -1, -7, 6], [5, -17, -1, 8], [-17, 1, -28, -4], [1, -19, -14, -12], [6, -3, -23, -15], [22, -14, -7, 12], [1, -7, 7, 14], [16, -4, -6, 4], [-17, -11, -15, -4], [-10, -15, -8, -19], [8, -23, -6, -7], [-51, -14, 12, -9], [-34, 6, 6, -8], [-10, 26, 14, 8], [1, 8, 11, -11], [-43, -11, -5, -12], [16, -3, -11, 11], [3, -1, -9, -14], [48, 2, -8, 6], [-25, -7, 3, -14], [-23, 3, 13, -6], [-48, 2, -2, -12], [12, -18, -22, 10], [-5, 15, 23, 4], [6, 12, -2, -10], [2, -2, -4, -8], [4, 14, 2, -12], [3, -1, -15, -14], [-19, -11, 21, 2], [4, -15, -12, -21], [-27, 15, 20, 16], [-22, 2, 4, 10], [-24, -10, -12, -12], [6, -2, 4, 12], [-10, -24, -10, 14], [30, -6, -10, -8], [-7, 11, 22, 10], [-15, -7, 3, -8], [-15, 21, -15, 2], [-8, -14, 28, 4], [-4, -6, 12, 14], [-36, -2, 11, 6], [3, 19, 11, 6], [-3, -1, 25, 16], [14, -4, -6, -12], [7, -9, 13, 0], [5, -8, 22, -3], [54, 12, 4, 6], [-8, -31, 9, -13], [29, 4, -1, 23], [10, 7, -1, 15], [-17, -10, 3, -3], [-14, -18, 26, 2], [-25, -10, -20, -1], [-20, 26, 2, 10], [-33, 1, 23, -6], [29, -9, -2, 6], [30, 28, -10, -16], [-14, -10, 10, -2], [-58, 0, 12, -14], [15, -22, -20, -13], [-17, 13, 5, -4], [-3, 16, 22, 21], [50, 6, 8, -6], [-31, -4, 8, 17], [-25, 23, 21, 8], [-26, 2, 0, 8], [-9, 20, 12, -3], [0, 2, 26, 10], [1, -28, -23, -3], [-37, -1, 14, 2], [50, -14, -14, -8], [17, 25, 17, -6], [-7, -15, -29, -8], [13, 1, 5, 6], [-26, 13, 12, 1], [11, 13, -29, 2], [-18, 1, 7, -13], [-7, -22, 15, 1], [7, 15, -31, 0], [20, 14, 19, 12], [16, 12, 12, 26], [-8, 4, -4, 12], [-16, -6, -22, -24], [-20, 11, 18, 7], [18, 24, -8, -16], [-2, 8, -28, -6], [-27, -4, -36, -7], [27, 9, -5, 6], [-5, 3, -4, 2], [2, -5, -9, 11], [-16, 0, 11, 10], [-25, -3, 18, 6], [2, 20, 23, 32], [-16, -6, -12, -24], [3, 19, 9, 12], [37, 11, -5, 8], [25, 15, -13, -2], [-33, -24, 2, -13], [0, -28, 1, -12], [-14, 30, 8, 14], [2, 16, 8, 24], [-22, -12, 4, -8], [-18, 14, -1, 12], [-44, 9, 3, 1], [41, -2, -4, -1], [50, 22, 0, -12], [-34, 8, 15, 10], [8, 4, 6, -2], [-7, -25, -15, -2], [-4, -24, 16, 0], [-26, 2, -12, -18], [-22, -4, -2, -6], [14, -10, -24, -8], [6, 25, 12, 13], [-2, 2, 6, 10], [36, 4, -4, 20], [-52, 24, 14, -2], [24, 23, 14, -11], [-36, 2, 16, -14], [22, 6, 22, 6], [-16, -20, 18, -6], [-60, 20, 9, -4], [-23, 15, 32, 4], [9, 10, -17, -17], [51, 9, -13, -6], [3, 7, -3, -18], [-44, 21, 7, 9], [11, -27, -1, -14], [-17, -2, 18, 3], [48, 18, -14, -22], [0, -14, -1, 10], [-34, 4, 6, 0], [10, 20, 24, 26], [28, -30, -26, 8], [-18, -12, -7, 8], [11, -9, -9, 0], [13, 2, -24, -5], [8, -14, -22, -12], [55, -10, -4, 9], [-33, -7, 17, -10], [16, -14, -6, -4], [-28, 23, 18, -13], [-1, 8, 18, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4235_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4235_2_a_u();".
// 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_4235_2_a_u(:prec:=4)
    chi := MakeCharacter_4235_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_4235_2_a_u();".
// 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_4235_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4235_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 2, -6, 0, 1]>,<3,R![-6, -14, -6, 2, 1]>,<13,R![-442, -14, 70, 16, 1]>],Snew);
    return Vf;
end function;