
// Make newform 231.2.a.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_231_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_231_2_a_c();".
// 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_231_2_a_c();".
// 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];
        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_231_a();"
function MakeCharacter_231_a()
    N := 231;
    order := 1;
    char_gens := [155, 199, 211];
    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_231_a_Hecke(Kf)
    return MakeCharacter_231_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], [1, 0], [1, 0], [1, 0], [1, 0], [1, -4], [4, -2], [-3, 6], [2, -6], [5, 0], [-4, 2], [-7, 0], [0, 4], [2, -6], [-1, -2], [-6, 10], [-5, 10], [2, 0], [-11, -2], [0, 4], [7, 4], [-12, 4], [8, 2], [-2, 4], [6, -6], [12, -10], [10, -12], [-3, 2], [4, 2], [2, 4], [-4, 14], [-4, -8], [16, -6], [12, -4], [5, 0], [-8, 10], [-8, 12], [-19, 6], [-14, 14], [-4, -4], [4, -8], [-2, -2], [-12, -2], [6, -14], [-14, 4], [-10, -10], [-2, -12], [-6, 0], [-2, 0], [6, -2], [-10, 8], [-1, -18], [21, -8], [1, -18], [-7, 0], [-15, 18], [6, 8], [9, 6], [-4, 4], [9, -4], [-15, -2], [-16, 0], [-16, 8], [0, -16], [16, -14], [18, -10], [-4, 12], [-2, 10], [28, 0], [9, 12], [27, 4], [16, -12], [-10, 16], [-4, -14], [-25, 10], [28, -8], [16, 8], [12, -18], [-26, 6], [26, 8], [5, 10], [-13, 0], [5, -6], [-8, 4], [5, -10], [16, -4], [20, 0], [14, -2], [-28, 0], [-11, -10], [23, -10], [8, -6], [12, -8], [-7, 18], [-1, 2], [6, 6], [-30, 0], [21, -28], [-13, 14], [-32, 8], [0, -24], [-15, -4], [26, -14], [30, 0], [20, 4], [16, 14], [25, -34], [-8, 24], [-28, -4], [27, 0], [13, 10], [0, -22], [30, -4], [-28, 26], [32, -20], [-28, 30], [-22, 42], [21, 14], [-40, 8], [5, -30], [22, -30], [-10, -22], [2, -28], [16, -34], [2, 0], [14, 16], [-25, 40], [9, 2], [18, -10], [-18, -16], [18, -26], [9, 10], [41, 2], [-27, 40], [8, -32], [9, -8], [-3, 4], [-5, -14], [-5, -4], [-39, 28], [19, 6], [-5, 4], [-11, 10], [21, 14], [16, -32], [-5, 30], [-22, 12], [10, 16], [-40, 0], [-42, 2], [12, 12], [45, -16], [-21, -10], [12, 2], [28, 0], [-22, -12], [26, -22], [3, -16], [-30, 16], [-48, 20], [2, 12], [25, -32], [-34, 44], [21, -18], [-2, -10], [20, -12], [-23, 10], [22, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_231_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_231_2_a_c();".
// 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_231_2_a_c(:prec:=2)
    chi := MakeCharacter_231_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(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_231_2_a_c();".
// 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_231_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_231_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 1]>],Snew);
    return Vf;
end function;