
// Make newform 931.2.v.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_931_v();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_931_v_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_931_2_v_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_931_2_v_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, 0, 0, -1, 0, 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_v();"
function MakeCharacter_931_v()
    N := 931;
    order := 9;
    char_gens := [248, 344];
    v := [3, 7];
    // 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_931_v_Hecke();"
function MakeCharacter_931_v_Hecke(Kf)
    N := 931;
    order := 9;
    char_gens := [248, 344];
    char_values := [[-1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, -1]];
    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 := [[1, 0, -1, -1, -1, 0], [-1, 0, 1, 0, 1, -1], [-1, -1, 0, 1, 0, -1], [0, 0, 0, 0, 0, 0], [0, -1, 2, 0, -1, -1], [2, 2, 0, -3, 1, 0], [1, 2, 1, 0, 0, 0], [-2, 4, 2, 2, -2, -1], [-2, 0, -2, 0, -2, 0], [-1, 1, -5, 1, -1, 0], [3, 1, 1, 0, -3, 2], [0, 3, -1, 0, -2, -2], [-4, -4, -3, 1, 0, -1], [2, 0, -2, -5, 0, -3], [1, 1, -2, 2, -3, 2], [-3, -3, 2, 2, 1, -2], [0, -7, 2, 2, 0, -2], [0, 4, -3, 4, 0, 0], [-2, -4, 0, 6, 6, -6], [-2, 0, 2, 2, -10, 0], [4, 0, -4, -4, 0, 0], [7, -3, 6, -6, 3, -7], [0, 9, -3, 0, -6, -6], [-3, 2, 0, 1, 1, -5], [-2, -5, 2, -2, 5, 2], [-1, -3, 5, -3, -1, 0], [6, 4, 4, 0, -3, -1], [0, -3, 6, 0, 6, -3], [2, 5, 4, -4, -5, -2], [-3, 1, 1, 0, 7, -8], [7, 7, 1, 1, -8, -1], [-1, 0, 1, 7, 7, 6], [1, 1, -10, -6, 5, 10], [-5, -5, 6, 5, 0, -6], [3, 8, -4, 4, -8, -3], [0, -3, 3, 10, 3, -3], [-8, 3, -5, 5, -3, 8], [0, -3, -3, 0, -2, 5], [8, -8, 0, 0, 0, -11], [-2, 0, 2, -12, -2, -14], [9, -3, -6, -9, 9, 9], [-10, 2, -2, 2, -10, 0], [-9, 7, 4, 9, -11, -11], [3, 0, -3, 2, 9, 5], [6, 10, -11, -6, 1, 1], [-8, 0, 0, 8, 8, 12], [-7, 3, -1, 1, -3, 7], [-3, 0, 3, 4, 9, 1], [0, 3, 6, -6, 6, 3], [3, 14, -6, -3, -8, -8], [9, 1, -5, -14, 0, 14], [0, -11, 1, 3, 1, -11], [-5, 2, -9, 2, -5, 0], [-3, -7, 0, 10, 10, -2], [-2, -2, 7, 5, -3, -7], [4, 0, -4, -13, -1, -9], [10, 0, -10, 3, 4, 13], [1, -7, -13, -14, 0, 14], [-5, 9, 3, 5, -12, -12], [-10, -3, 0, 13, 13, -4], [-8, -7, -1, 7, 0, -7], [0, -8, 7, 3, 7, -8], [13, 11, 5, -8, 0, 8], [-9, 7, 7, 0, -2, -5], [-10, -10, -10, 17, -7, 10], [-1, 0, 0, 1, 1, -28], [15, 4, 4, 0, -10, 6], [5, 0, -5, -8, 15, -3], [-3, -1, 2, -2, 1, 3], [0, -6, -11, 0, 17, 17], [-15, 6, -3, 15, -3, -3], [-1, 0, 1, 6, 14, 5], [-14, 8, 9, 8, -14, 0], [0, 6, 15, -14, 15, 6], [15, 5, 5, 0, 3, -8], [5, 16, 11, -11, -16, -5], [9, -8, 0, -1, -1, 8], [-13, 0, 13, 3, 12, -10], [-3, 0, 3, 7, 5, 4], [14, -16, 0, 2, 2, -7], [-3, -5, -5, 0, 16, -11], [2, -2, 4, 2, 0, -2], [-4, 13, -17, 13, -4, 0], [-8, -9, 0, 17, 17, -9], [-1, 0, 1, 9, 18, 8], [20, -18, 0, -2, -2, -7], [-15, -16, 5, 15, 11, 11], [0, -13, 10, 0, 3, 3], [-7, 0, 7, 1, 25, -6], [15, -16, 10, -15, 6, 6], [-21, 3, 3, 0, 0, -3], [1, 1, 4, -19, 18, -4], [-1, 9, 9, 0, -12, 3], [-16, 7, -1, 15, 0, -15], [2, 3, -12, -14, 0, 14], [-17, 17, -1, 17, -17, 0], [21, -20, -1, -20, 21, 0], [-18, -16, 8, 18, 8, 8], [21, 21, 0, -2, -19, 0], [-1, -1, -9, 4, -3, 9], [6, -2, 0, -4, -4, -6], [5, 1, 13, 1, 5, 0], [9, 19, 19, 0, -5, -14], [12, -3, -9, -12, 12, 12], [0, -10, 6, 0, 6, -10], [7, -6, -6, 0, 3, 3], [7, 28, 6, -6, -28, -7], [2, 0, 8, 0, 2, 0], [-22, 9, 0, 13, 13, 17], [-27, 6, 7, 27, -13, -13], [-6, -24, 13, 6, 11, 11], [2, 14, -2, 2, -14, -2], [12, 11, -23, 11, 12, 0], [18, 8, -6, -18, -2, -2], [10, -10, 0, 0, 0, -18], [1, 14, -8, 8, -14, -1], [5, 33, -4, -9, 0, 9], [0, 19, -5, -3, -5, 19], [0, 0, 0, -27, 0, 0], [25, 25, -1, -6, -19, 1], [2, 2, -10, -13, 11, 10], [0, 8, -9, 0, -9, 8], [0, -4, 5, 0, 5, -4], [0, 5, -1, -12, -1, 5], [0, 24, -12, 8, -12, 24], [-21, 11, -19, 19, -11, 21], [12, -5, 0, -7, -7, 21], [-29, 13, -21, 21, -13, 29], [-4, 0, 4, 21, -27, 17], [25, -6, -6, 0, 6, 0], [13, -3, -2, -15, 0, 15], [12, 17, 8, -8, -17, -12], [13, 13, 3, -6, -7, -3], [-4, -18, -15, -11, 0, 11], [-3, -4, 5, 3, -1, -1], [0, 1, -21, 1, 0, 0], [13, -13, 2, -13, 13, 0], [12, 13, -22, -12, 9, 9], [-9, -11, -11, 0, 22, -11], [-27, 4, -23, 27, 19, 19], [-1, -1, 2, -7, 8, -2], [-10, 13, -8, 13, -10, 0], [10, -11, 26, -11, 10, 0], [2, -2, 37, -2, 2, 0], [21, -7, -7, 0, -2, 9], [8, -19, -12, -20, 0, 20], [-29, 15, -11, 11, -15, 29], [4, -16, 7, 3, 0, -3], [8, -9, 3, -9, 8, 0], [-6, -28, -28, 0, 17, 11], [-2, -15, -9, 9, 15, 2], [33, -7, 8, -33, -1, -1], [1, -18, -15, -18, 1, 0], [-4, -17, 0, 21, 21, -23], [23, -1, 25, -25, 1, -23], [-36, -10, -10, 0, -1, 11], [0, -9, 22, -3, 22, -9], [-7, 0, 7, 19, -11, 12], [1, 1, -9, -7, 6, 9], [6, 14, 0, -20, -20, -23], [-13, -4, -9, 9, 4, 13], [9, -14, -8, -17, 0, 17], [8, 24, 8, -8, -24, -8], [12, 0, -12, 8, 22, 20], [3, 6, -24, -3, 18, 18], [27, 7, -11, 11, -7, -27], [21, 0, -21, -28, 0, -7], [3, 11, 0, -14, -14, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_931_v_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_931_2_v_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_931_2_v_a(:prec:=6)
    chi := MakeCharacter_931_v();
    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_v_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_931_2_v_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_931_v();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -27, 36, -24, 9, -3, 1]>,<3,R![1, 3, 6, 8, 12, 6, 1]>],Snew);
    return Vf;
end function;