
// Make newform 931.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_931_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_931_2_a_f();".
// 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_931_2_a_f();".
// 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 := [-2, 0, 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_931_a();"
function MakeCharacter_931_a()
    N := 931;
    order := 1;
    char_gens := [248, 344];
    v := [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_931_a_Hecke(Kf)
    return MakeCharacter_931_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, 1], [0, -1], [1, 0], [0, 0], [1, 1], [2, 3], [-4, 0], [-1, 0], [-7, -1], [-8, -1], [-2, 3], [2, -5], [-6, -2], [1, -5], [-3, 1], [0, 3], [4, -2], [9, -2], [4, 2], [-10, 3], [1, -10], [0, -1], [5, -3], [-10, 1], [-6, -2], [7, -8], [-6, 0], [-4, 7], [-4, 4], [4, 3], [0, -3], [12, 6], [-17, 2], [1, 5], [-3, -2], [-6, 8], [9, 2], [11, 5], [6, -5], [4, 8], [-2, -14], [0, -7], [7, -1], [-2, -7], [13, 6], [9, -5], [-4, -2], [-2, -3], [-16, 3], [4, -6], [-8, -6], [2, 0], [-2, 17], [-1, 9], [-2, -8], [8, -6], [-4, 0], [-11, 3], [3, 18], [-24, 3], [23, -5], [6, 2], [18, -8], [-4, 14], [-1, 4], [-16, 3], [-6, 18], [4, -11], [3, -17], [-4, -6], [-8, -6], [1, -3], [-30, -4], [0, -8], [0, -3], [24, 3], [-20, -6], [14, 16], [-8, -6], [-14, -8], [-1, -5], [-16, 15], [14, 17], [28, 0], [-16, 6], [12, -4], [4, -4], [-5, 2], [13, -12], [3, 5], [29, 7], [-7, 13], [18, 2], [-17, -11], [19, -11], [-5, 17], [2, 12], [26, -11], [-18, -4], [-9, 10], [14, 11], [13, -2], [-4, -7], [-26, -12], [-9, -7], [11, 12], [14, 0], [9, -2], [-2, -21], [-22, 9], [0, 23], [-20, 8], [-17, -18], [19, -11], [-21, 11], [-16, 14], [16, 10], [-15, 15], [-16, 10], [-14, -17], [-10, -9], [16, 14], [12, -8], [2, 6], [14, -12], [1, 2], [-9, 24], [4, -34], [13, -5], [-4, -18], [-14, -4], [24, 19], [-16, -18], [-5, 0], [-3, 22], [-1, 30], [-30, 3], [0, -29], [-26, 3], [17, -10], [-26, -18], [-21, -22], [15, 15], [-16, -20], [0, 20], [0, 14], [27, -12], [4, 12], [-5, 5], [6, 1], [-18, 7], [-24, 20], [0, 4], [-10, 8], [30, -1], [22, 9], [5, 17], [15, 0], [-45, -8], [-20, -17], [-2, -4], [-2, 1], [-8, 20], [0, -19], [-8, 26], [-46, 0], [-16, -14], [0, 20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_931_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_931_2_a_f();".
// 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_931_2_a_f(:prec:=2)
    chi := MakeCharacter_931_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_931_2_a_f();".
// 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_931_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_931_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 1]>,<3,R![-2, 0, 1]>,<5,R![-1, 1]>,<13,R![-14, -4, 1]>],Snew);
    return Vf;
end function;