
// Make newform 23.2.c.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_23_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_23_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_23_2_c_a();".
// 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_23_2_c_a();".
// 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, 1, -1, 1, -1, 1, -1, 1, -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_23_c();"
function MakeCharacter_23_c()
    N := 23;
    order := 11;
    char_gens := [5];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_23_c_Hecke();"
function MakeCharacter_23_c_Hecke(Kf)
    N := 23;
    order := 11;
    char_gens := [5];
    char_values := [[0, -1, 0, 0, 0, 0, 0, 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, -1, 0, 0, 1, -1, 1, -1, 1, -1], [-1, 1, 0, 0, -1, 1, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 1, -2, 1, 0, 1], [0, -1, 1, -1, -1, -1, 1, -1, 0, 0], [3, -2, 2, -3, 3, -3, 3, -2, 2, -3], [-1, 1, -1, 1, 0, 0, -2, 0, -2, 0], [-1, -1, 0, -1, 1, 0, 1, 1, 0, 3], [0, 0, -2, 0, 0, 0, -2, -2, 2, 2], [-2, 0, -1, -1, -3, 2, -1, 3, 0, -1], [2, -1, 2, 2, -2, -2, 1, -2, 0, -2], [2, 0, 2, -2, 2, 0, 2, 0, 1, -1], [0, -1, 4, -2, 1, -3, 1, -2, 4, -1], [0, 0, 0, 3, -1, 0, 1, 0, -1, 3], [0, -3, 1, -1, 3, 0, 0, -1, 1, -1], [-2, 0, 1, 2, 1, 1, -1, -1, -2, -1], [1, 2, -6, 1, -2, 1, -6, 2, 1, 0], [-2, 0, 0, 2, 0, 0, 3, 3, 3, 0], [-3, 5, -6, 4, -6, 5, -3, 0, -2, 2], [7, -6, 7, 0, 1, -3, 2, -2, 3, -1], [-3, 3, -3, 0, -1, 1, -3, 3, -1, 1], [4, -2, -1, -2, 4, 0, 4, -3, 3, -4], [-5, 2, -2, 5, 0, 2, -8, 6, -8, 2], [0, 3, -3, -1, -2, 4, -2, -1, -3, 3], [-3, 7, -4, 4, -7, 3, 0, 11, -8, 11], [0, 0, 0, -5, 2, -7, 6, -7, 2, -5], [0, -7, 4, -8, 2, 1, 2, -8, 4, -7], [1, -6, -4, -3, 4, -4, 3, 4, 6, -1], [3, 1, 5, 1, 5, 1, 3, 0, 0, 0], [0, 3, 7, 1, -1, -7, -3, 0, 0, -5], [1, 0, 1, 0, -1, 1, -1, 1, -1, 1], [-3, 5, -11, 5, -10, 10, -5, 11, -5, 3], [-6, 0, -3, 2, -9, 2, -3, 0, -6, 0], [1, 0, 2, -2, -2, 4, -4, 2, 2, -2], [8, 0, 5, -14, 7, -6, 6, -7, 14, -5], [2, -6, 6, 1, 6, -6, -1, -6, 6, -2], [6, -4, 4, -6, 0, 3, 5, -4, 5, 3], [2, 3, 0, 3, 2, 0, -2, 8, -8, 2], [2, -4, 2, 0, -1, 8, 3, -3, -8, 1], [0, -1, 3, -5, 5, -3, 1, 0, 0, -2], [4, -3, 3, 2, 7, -7, -2, -3, 3, -4], [5, -5, 0, -9, 6, -10, 12, -10, 6, -9], [-11, 1, 1, -1, -1, 11, 0, 0, -4, 0], [-2, -3, 5, -8, -8, -8, 5, -3, -2, 0], [0, -2, 1, 5, 4, -2, 4, 5, 1, -2], [2, 3, -3, -2, 0, 4, -6, 5, -6, 4], [-5, 0, 2, 1, 2, 0, -5, 0, 0, 0], [-1, 3, -6, 3, -1, 0, -8, 2, -2, 8], [6, 3, 6, 6, -2, 6, 6, 3, 6, 0], [2, -6, -10, 10, 6, -2, 0, -5, -5, -5], [2, 0, -5, -1, 3, -5, 5, -3, 1, 5], [13, -6, 19, -19, 6, -13, 0, -10, 11, -10], [0, 9, -7, 5, -10, 14, -10, 5, -7, 9], [5, 7, 1, 8, -5, 5, -8, -1, -7, -5], [-1, 18, -1, 0, 1, -6, -4, 4, 6, -1], [-14, 4, 3, 4, -14, 0, -6, 5, -5, 6], [-8, 8, -8, 8, 0, 3, -1, 16, -1, 3], [-6, -1, 5, -4, 10, -4, 5, -1, -6, 0], [0, 0, 0, 10, -14, 3, -15, 3, -14, 10], [-14, 0, 0, 2, -9, 0, 0, 9, -2, 0], [-11, 11, 0, 0, 9, 1, -1, 1, 9, 0], [12, -5, 7, -11, 7, -11, 7, -5, 12, 0], [-2, -10, -4, 10, -10, 4, 10, 2, 0, -3], [-1, 8, 3, 10, 3, 8, -1, 0, -8, 8], [10, -6, 2, -11, 4, -4, 11, -2, 6, -10], [0, 12, -10, -1, -1, -9, -1, -1, -10, 12], [10, -10, 0, -1, 2, -6, 9, -6, 2, -1], [0, -9, -3, 4, 4, -11, 4, 4, -3, -9], [7, -2, -3, -4, -3, -2, 7, 0, 9, -9], [-9, 0, -9, 0, -2, 3, 2, -2, -3, 2], [-4, -10, 9, -10, -4, 0, -3, 1, -1, 3], [-17, 11, -17, 7, -17, 11, -17, 0, -15, 15], [0, -6, 5, -2, 7, 8, 7, -2, 5, -6], [-1, 0, 2, -6, 13, 5, -5, -13, 6, -2], [-9, 9, 0, 16, -8, 9, -2, 9, -8, 16], [-15, 4, -9, 6, 0, 0, -6, 9, -4, 15], [-12, 11, -3, 17, -3, 11, -12, 0, -4, 4], [-6, 7, -6, 0, -7, 16, -6, 6, -16, 7], [6, -8, 7, -16, 16, -7, 8, -6, 0, -1], [8, -10, 10, -8, 0, -16, 14, -13, 14, -16], [8, -8, 0, -8, 5, 8, 2, 8, 5, -8], [16, -16, 0, -14, -1, -2, -1, -2, -1, -14], [5, -13, 5, 3, 21, 3, 5, -13, 5, 0], [8, -7, 8, -13, 13, -8, 7, -8, 0, 9], [-1, -1, 15, -1, -1, 0, 7, -12, 12, -7], [-2, 4, -2, 0, 2, -12, 0, 0, 12, -2], [-9, 14, -11, 16, -16, 11, -14, 9, 0, 22], [-6, 8, -3, -3, -8, 8, 3, 3, -8, 6], [13, -11, 11, -11, 11, -13, 0, -3, 24, -3], [-11, 0, -14, 7, -14, 17, -17, 14, -7, 14], [-10, 10, -6, 6, -10, 10, 0, -9, -8, -9], [-4, 18, -3, 9, -4, 9, -3, 18, -4, 0], [9, 0, -14, 0, 9, 0, 5, -2, 2, -5], [5, 7, 8, 7, 5, 0, 7, -14, 14, -7], [13, -17, 8, 2, 8, -17, 13, 0, 5, -5], [-17, 12, -21, 13, -14, 13, -21, 12, -17, 0], [13, -3, 4, -4, 3, -13, 0, -3, -3, -3], [16, -14, 7, -7, 14, -16, 0, -6, 15, -6], [-9, -5, 12, -7, 12, -5, -9, 0, 4, -4], [11, -18, 17, -19, 19, -17, 18, -11, 0, -26], [-20, 11, -12, 18, -11, 11, -18, 12, -11, 20], [-1, 1, 0, -8, -4, -10, 17, -10, -4, -8], [12, -12, 0, -6, 0, -8, 7, -8, 0, -6], [-12, -6, -5, -5, -3, 3, 5, 5, 6, 12], [4, 4, -6, -17, 17, 6, -4, -4, 0, -2], [-19, 20, -16, 20, -19, 0, -8, 10, -10, 8], [0, -8, 0, 0, -6, -5, -12, 12, 5, 6], [11, 1, 22, -18, 12, -12, 18, -22, -1, -11], [1, -1, 0, -3, 8, 5, -13, 5, 8, -3], [1, 0, -7, 6, -8, 16, -16, 8, -6, 7], [-7, 9, -13, 13, -9, 7, 0, 8, -8, 8], [0, -7, 5, -9, 12, -8, 12, -9, 5, -7], [3, -11, -8, -7, -8, -11, 3, 0, 18, -18], [10, -1, 24, -1, 10, 0, 3, -5, 5, -3], [11, -7, 11, 0, 16, -3, 9, -9, 3, -16], [3, 15, -15, -3, 0, 0, -19, -4, -19, 0], [-12, -5, 1, -1, 5, 12, 0, -4, -5, -4], [10, 0, 23, -11, 9, -10, 10, -9, 11, -23], [-8, 26, -7, 7, -26, 8, 0, 15, -8, 15], [0, 13, -8, 7, 8, 12, 8, 7, -8, 13], [6, -17, 8, 5, 8, -17, 6, 0, 4, -4], [-20, 20, -16, 23, -23, 16, -20, 20, 0, 15], [13, 4, 9, -3, 3, -9, -4, -13, 0, 17], [14, -14, 14, -14, 0, -22, 7, -14, 7, -22], [14, -3, 7, -4, 23, -4, 7, -3, 14, 0], [0, 0, 11, -3, -7, 7, -7, 7, 3, -11], [15, -9, 5, -3, 9, -9, 3, -5, 9, -15], [-2, -7, 5, -7, -2, 0, -3, 4, -4, 3], [-5, -14, 5, 3, -3, -5, 14, 5, 0, 7], [0, -8, -1, -5, 22, 2, 22, -5, -1, -8], [2, 12, 7, -7, -12, -2, 0, -7, 3, -7], [-5, -8, 11, -11, 8, 5, 0, -4, 4, -4], [-13, 3, -27, 20, -15, 20, -27, 3, -13, 0], [9, 14, 1, 15, 1, 14, 9, 0, -7, 7], [-14, 7, -14, 0, -12, 7, -19, 19, -7, 12], [11, -1, 11, 0, 6, 2, 20, -20, -2, -6], [10, -9, 9, -10, 0, -1, 28, 3, 28, -1], [0, 16, -3, 0, -11, 8, -11, 0, -3, 16], [-20, 20, 0, 8, -3, 6, -11, 6, -3, 8], [-23, 11, -26, 18, -26, 11, -23, 0, -23, 23], [-19, 12, -21, 12, -19, 0, -7, 13, -13, 7], [7, -1, 10, -19, 19, -10, 1, -7, 0, -35], [1, 15, -16, 1, -6, 1, -16, 15, 1, 0], [-14, 14, 0, 10, -5, 26, -11, 26, -5, 10], [-11, 0, -33, 16, -14, 15, -15, 14, -16, 33], [-6, 0, 6, -1, 14, -7, 7, -14, 1, -6], [21, -7, 28, -30, 12, -12, 30, -28, 7, -21], [-22, 17, -22, 0, -14, 22, -16, 16, -22, 14], [10, -4, -21, 1, -1, 21, 4, -10, 0, 1], [-11, 10, -24, 30, -24, 10, -11, 0, -17, 17], [-16, 7, -12, 13, -20, 20, -13, 12, -7, 16], [-1, 3, -8, 8, -3, 1, 0, 6, -4, 6], [-4, -2, 11, -5, -8, -5, 11, -2, -4, 0], [0, -9, 20, 0, 1, 1, 1, 0, 20, -9], [18, -18, 18, -18, 0, -1, 0, -16, 0, -1], [6, -21, 21, -6, 0, -5, 6, -6, 6, -5], [0, -11, 19, -10, 15, 10, 15, -10, 19, -11], [-16, 0, -4, 20, -2, -13, 13, 2, -20, 4], [0, 9, -5, 12, -9, 24, -9, 12, -5, 9], [13, -3, -4, 6, -6, 4, 3, -13, 0, 2], [17, -4, 17, 0, 13, -22, 27, -27, 22, -13], [2, -14, 3, -14, 2, 0, 22, -20, 20, -22], [-14, 5, -5, 14, 0, 2, -22, 11, -22, 2], [0, 0, -18, 8, 1, 3, -3, -1, -8, 18], [-16, 16, 0, 9, -32, 26, -23, 26, -32, 9], [-21, 18, -12, 21, 6, -6, -21, 12, -18, 21], [23, 7, -5, 0, 0, 5, -7, -23, 0, 0], [-15, 20, -15, 0, -38, 10, -14, 14, -10, 38], [3, -13, 9, -4, 9, -13, 3, 0, 14, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_23_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_23_2_c_a();".
// 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_23_2_c_a(:prec:=10)
    chi := MakeCharacter_23_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(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_23_2_c_a();".
// 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_23_2_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_23_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;