
// Make newform 4032.2.c.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4032_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4032_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_c_p();".
// 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_4032_2_c_p();".
// 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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [0, 0, 0, 2], [-1, 0, 2, -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_4032_c();"
function MakeCharacter_4032_c()
    N := 4032;
    order := 2;
    char_gens := [127, 3781, 1793, 577];
    v := [2, 1, 2, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4032_c_Hecke();"
function MakeCharacter_4032_c_Hecke(Kf)
    N := 4032;
    order := 2;
    char_gens := [127, 3781, 1793, 577];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [1, 0, 0, 0], [0, 0, -1, -3], [0, 0, 0, 0], [3, -1, 0, 0], [0, 0, 3, 2], [7, 1, 0, 0], [0, 0, 1, 2], [2, -4, 0, 0], [0, 0, 5, 0], [-3, 1, 0, 0], [0, 0, -1, 4], [0, -4, 0, 0], [0, 0, -1, -2], [0, 0, -4, -4], [0, 0, 0, -2], [0, 0, -3, -2], [9, 3, 0, 0], [0, 6, 0, 0], [6, 2, 0, 0], [0, 0, -2, -6], [3, -1, 0, 0], [-4, 6, 0, 0], [0, 0, 2, 7], [2, 4, 0, 0], [0, 0, -3, -1], [0, 0, 0, 4], [-12, 2, 0, 0], [10, -6, 0, 0], [0, 0, -8, -6], [4, 6, 0, 0], [0, 0, 0, -4], [0, 0, 7, -2], [-8, -4, 0, 0], [0, 0, 0, 2], [0, 0, -1, 6], [-2, -2, 0, 0], [0, 0, 0, 5], [0, 0, 5, -3], [0, 0, -6, 4], [-11, -1, 0, 0], [10, 2, 0, 0], [0, 0, 11, 4], [-8, -8, 0, 0], [0, 0, -5, -8], [4, 12, 0, 0], [0, 0, 6, 2], [0, 0, 10, 12], [-2, 8, 0, 0], [-7, 11, 0, 0], [0, -14, 0, 0], [0, 0, -8, 2], [5, 9, 0, 0], [5, 11, 0, 0], [0, 0, 6, 9], [14, 8, 0, 0], [0, 0, -7, -12], [2, 8, 0, 0], [0, 0, -7, -2], [0, 0, -4, -11], [0, 0, -3, -2], [16, -4, 0, 0], [-10, 4, 0, 0], [0, 0, -9, -10], [0, 0, -13, -4], [2, 8, 0, 0], [0, 0, 7, 5], [0, 0, 10, 2], [-17, -9, 0, 0], [-1, -19, 0, 0], [24, -8, 0, 0], [0, 0, 2, 0], [0, 0, -7, -4], [-18, -6, 0, 0], [0, 0, 7, 2], [0, 0, -2, 10], [-8, 6, 0, 0], [28, 6, 0, 0], [0, 0, 6, 8], [0, 0, -1, -12], [-11, 11, 0, 0], [-18, -4, 0, 0], [0, -8, 0, 0], [0, 0, -3, 1], [12, -2, 0, 0], [-6, 4, 0, 0], [0, 0, -6, -9], [-2, -14, 0, 0], [0, 0, 0, 10], [-8, -16, 0, 0], [8, 8, 0, 0], [0, 0, 13, 11], [0, 0, 7, -8], [26, -6, 0, 0], [0, 0, -20, -1], [-27, 1, 0, 0], [0, 0, 14, 20], [0, 0, -3, 8], [0, 0, 1, 6], [0, 0, -11, -10], [0, 0, -8, -8], [-10, -4, 0, 0], [0, 0, 3, 2], [2, -12, 0, 0], [0, 0, 2, -2], [5, -23, 0, 0], [17, -5, 0, 0], [14, -16, 0, 0], [-28, 4, 0, 0], [0, 0, 14, 0], [-12, -14, 0, 0], [0, 0, -2, 0], [-30, 2, 0, 0], [-36, 6, 0, 0], [0, 0, 1, 6], [-22, -6, 0, 0], [0, 0, 5, 12], [0, 0, -1, -11], [0, 0, 10, 2], [-14, 10, 0, 0], [0, 0, -20, -9], [0, 0, -11, -7], [0, 0, -4, 4], [0, 0, -11, -4], [0, 0, 4, 12], [30, 2, 0, 0], [14, 16, 0, 0], [0, 0, 4, 4], [0, 0, 7, 18], [-25, 13, 0, 0], [20, 4, 0, 0], [0, 0, -20, 0], [27, -5, 0, 0], [-14, 8, 0, 0], [0, 0, -10, 11], [0, 0, -12, -4], [0, 0, -16, -3], [16, 10, 0, 0], [0, 0, 14, 20], [0, 0, 23, 10], [-2, 14, 0, 0], [0, 0, 25, 5], [0, 0, -6, -4], [-6, 14, 0, 0], [0, 0, 6, 6], [-29, -13, 0, 0], [0, 0, -7, -14], [-29, 1, 0, 0], [0, 0, 2, 4], [25, -11, 0, 0], [0, 0, 15, 28], [-34, -2, 0, 0], [0, 0, -7, -8], [21, -5, 0, 0], [20, -4, 0, 0], [-33, 3, 0, 0], [46, -8, 0, 0], [0, 0, 6, 11], [0, 0, 3, 11], [-18, -12, 0, 0], [14, 14, 0, 0], [0, 0, 16, 28], [-14, -16, 0, 0], [-8, -28, 0, 0], [-32, 4, 0, 0], [0, 0, 10, -2], [10, -10, 0, 0], [0, 0, 6, 21], [0, 0, -3, 15], [0, 0, -22, 6], [9, -21, 0, 0], [-14, -10, 0, 0], [28, -12, 0, 0], [-17, 27, 0, 0], [0, 0, -7, -22], [0, 0, -17, -4], [-2, -28, 0, 0], [0, 0, 6, -6], [-14, -26, 0, 0], [0, 0, 22, 8], [0, 0, 19, -8], [27, -1, 0, 0], [33, 3, 0, 0], [0, 0, 22, 17], [0, 0, -12, 8], [0, 0, 8, 8], [18, -22, 0, 0], [-34, -18, 0, 0], [16, -2, 0, 0], [0, 0, 9, 9], [0, 0, 19, -4], [0, 0, 22, 7], [0, 0, 1, -5], [-41, -5, 0, 0], [38, -2, 0, 0], [0, 0, 13, 0], [31, 15, 0, 0], [16, -4, 0, 0], [0, 0, -1, -12], [-20, 12, 0, 0], [0, 0, 6, -20], [-16, 10, 0, 0], [0, 0, 22, 4], [0, 0, 8, 19], [8, 8, 0, 0], [0, 0, 23, 7], [22, 0, 0, 0], [0, 0, -7, -14], [-2, -40, 0, 0], [0, 0, -2, 1], [40, 12, 0, 0], [0, 0, 14, 4], [-32, 8, 0, 0], [-10, 4, 0, 0], [8, 12, 0, 0], [41, 9, 0, 0], [-15, -9, 0, 0], [0, 0, -15, -26], [0, 0, 16, 8], [-10, 0, 0, 0], [-16, 10, 0, 0], [-28, -4, 0, 0], [0, 0, -30, -6], [0, 0, -11, -12], [-15, 1, 0, 0], [-25, 9, 0, 0], [22, -12, 0, 0], [0, 0, 3, 1], [0, 0, 4, -8], [0, 0, -21, -2], [8, 24, 0, 0], [36, 10, 0, 0], [0, 0, 5, -10], [48, -8, 0, 0], [-8, 6, 0, 0], [0, 0, -13, 0], [0, 0, -27, -23], [-50, 6, 0, 0], [0, 0, 15, 15], [0, 0, 6, -12], [-36, 4, 0, 0], [0, 0, -5, 0], [-19, -15, 0, 0], [34, -2, 0, 0], [-30, -20, 0, 0], [0, 0, -2, 22], [0, 0, -25, -34], [41, -17, 0, 0], [0, 0, 4, 0], [-47, -11, 0, 0], [3, -3, 0, 0], [32, 22, 0, 0], [0, 0, 2, 9], [0, 0, -11, -31], [0, 0, -9, -8], [0, 0, -11, 18], [0, 0, 8, 25], [2, -32, 0, 0], [2, 26, 0, 0], [0, 0, 11, 9], [0, 0, 30, 8], [0, 0, 10, 6], [-51, 13, 0, 0], [0, 0, -6, -4], [0, 0, -11, -32], [37, -19, 0, 0], [0, 0, 31, 6], [0, 0, -13, -20], [0, 0, -18, -14], [0, 0, 19, -4], [40, 10, 0, 0], [8, 4, 0, 0], [32, 14, 0, 0], [-58, 0, 0, 0], [0, 0, -9, 3], [0, 0, 31, 16], [42, 2, 0, 0], [0, 0, -2, 2], [-20, -32, 0, 0], [-26, -14, 0, 0], [12, 0, 0, 0], [0, 0, -28, -30], [0, 0, 30, 16], [-7, 23, 0, 0], [66, -4, 0, 0], [0, 0, 1, -4], [-14, 8, 0, 0], [33, -3, 0, 0], [0, 0, -5, 0], [0, 0, 0, -18], [-16, -34, 0, 0], [0, 0, 8, 12], [0, 0, 10, 24], [0, 0, -4, -27], [-24, -24, 0, 0], [0, 0, 6, 27], [0, 0, -2, -10], [0, 0, 2, -8], [2, 20, 0, 0], [0, 0, 19, 6], [2, -22, 0, 0], [0, 0, -21, -13], [0, 0, -21, 2], [-62, -8, 0, 0], [0, 0, -29, -9], [0, 0, 36, 12], [51, 9, 0, 0], [0, 0, -23, -12], [0, 12, 0, 0], [0, 0, -23, -32], [-46, -4, 0, 0], [0, 0, 19, 4], [9, 7, 0, 0], [-16, -30, 0, 0], [0, 0, -20, -16], [9, -1, 0, 0], [12, -10, 0, 0], [-16, 34, 0, 0], [0, 0, -13, 10], [-34, -14, 0, 0], [0, 0, 16, -5], [-12, 8, 0, 0], [12, -30, 0, 0], [52, -6, 0, 0], [0, 0, -7, 18], [0, 0, -9, -26], [-9, -3, 0, 0], [0, 0, -15, -8], [0, 0, 16, -8], [0, 0, 13, 12], [38, 8, 0, 0], [0, 0, -26, -24], [0, 0, -17, -2], [0, 0, 22, 4], [0, 0, -6, -16], [29, 1, 0, 0], [-12, -10, 0, 0], [2, -28, 0, 0], [0, 0, -2, -24], [-14, 16, 0, 0], [0, 0, -2, -17], [-52, 8, 0, 0], [0, 0, -5, 18], [0, 0, -29, -13], [0, 0, 16, 24], [0, 0, -19, -28], [28, -12, 0, 0], [0, 0, 6, -17], [0, 0, -19, -38], [-10, 2, 0, 0], [0, 0, 3, -24], [-8, 0, 0, 0], [0, 0, 26, -12], [23, -21, 0, 0], [-42, 2, 0, 0], [0, 0, -32, 0], [-24, 22, 0, 0], [21, -29, 0, 0], [0, 0, -2, 28], [13, 21, 0, 0], [13, -37, 0, 0], [0, 0, -39, 3], [0, 0, 32, 16], [-46, -18, 0, 0], [0, 0, 26, 21], [18, -22, 0, 0], [-2, 18, 0, 0], [0, 0, 9, 27], [0, 0, 15, -10], [-11, 7, 0, 0], [0, 0, -9, -8], [58, -16, 0, 0], [0, 0, -23, -28], [0, 0, 28, 22], [-77, 5, 0, 0], [-10, 24, 0, 0], [9, 25, 0, 0], [60, 10, 0, 0], [0, 0, 32, -3], [48, 10, 0, 0], [-4, 0, 0, 0], [-66, 16, 0, 0], [0, 0, -30, -28], [-42, 14, 0, 0], [64, 12, 0, 0], [0, 0, 6, 12], [0, 0, 31, 30], [-48, -8, 0, 0], [-18, -6, 0, 0], [0, 0, -12, -5], [0, 0, -3, 3], [0, 0, -17, -18], [1, -9, 0, 0], [26, -28, 0, 0], [24, 0, 0, 0], [-57, -1, 0, 0], [0, 0, -21, -34], [0, 0, 11, 38], [0, 0, -12, 14], [4, 2, 0, 0], [14, -2, 0, 0], [29, -31, 0, 0], [0, 0, 26, 23], [-24, -16, 0, 0], [0, 0, 43, 24], [2, -24, 0, 0], [0, 0, 14, -4], [0, 0, -42, -12], [-34, 4, 0, 0], [0, 0, -13, 6], [0, 0, 3, 23], [0, 0, -39, -16], [10, -10, 0, 0], [0, 0, -26, -7], [-43, -5, 0, 0], [22, -8, 0, 0], [33, 25, 0, 0], [40, 0, 0, 0], [0, 0, -9, -38], [0, 0, -6, 30], [15, 41, 0, 0], [0, 0, 4, -34], [-34, -28, 0, 0], [0, 0, 28, 35], [0, 0, -25, 19], [10, 28, 0, 0], [0, 0, 36, 24], [42, 10, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4032_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_c_p();".
// 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_4032_2_c_p(:prec:=4)
    chi := MakeCharacter_4032_c();
    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_4032_2_c_p();".
// 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_4032_2_c_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 0, 8, 0, 1]>,<11,R![676, 0, 56, 0, 1]>,<13,R![0, 1]>,<17,R![6, -6, 1]>,<23,R![46, -14, 1]>,<31,R![-44, -4, 1]>],Snew);
    return Vf;
end function;