
// Make newform 114.2.i.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_114_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_114_i_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_114_2_i_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_114_2_i_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_114_i();"
function MakeCharacter_114_i()
    N := 114;
    order := 9;
    char_gens := [77, 97];
    v := [9, 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_114_i_Hecke();"
function MakeCharacter_114_i_Hecke(Kf)
    N := 114;
    order := 9;
    char_gens := [77, 97];
    char_values := [[1, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 0, 1, -1, 0, 1], [3, 0, -1, -3, 1, 1], [0, -2, 1, 0, 1, -2], [3, -2, 0, -1, -1, 0], [-2, 0, -1, 1, 0, -1], [-2, 1, 2, 0, -2, -4], [2, -1, 0, -1, 2, 0], [-2, -2, -6, 1, 1, 6], [-3, 5, 1, 3, -6, -6], [3, -2, -2, 0, 2, 0], [3, 0, -3, 3, 3, 6], [-3, -3, 1, -1, 3, 3], [4, 4, -3, 1, -5, 3], [2, -4, 3, -4, 2, 0], [0, -6, 3, 3, 0, -3], [-3, 5, 3, 5, -3, 0], [-2, -2, -6, 3, -1, 6], [3, 9, 3, -3, -9, -3], [-5, 0, 5, 6, -3, 1], [-5, 0, 5, 3, 9, -2], [-3, -7, 8, 3, -1, -1], [1, 1, 0, -2, -2, -9], [-9, 4, -9, 0, 0, 0], [-4, 8, 0, -4, -4, -9], [0, -8, -1, -3, -1, -8], [-3, 8, -1, 3, -7, -7], [0, -7, -3, 3, 7, 0], [0, -10, -10, 0, -1, 11], [6, 0, 0, -6, -6, 4], [7, 6, -1, -8, 0, 8], [2, 2, 9, 2, 2, 0], [-1, -3, 0, 4, 4, 9], [1, 0, -1, 1, 0, 2], [2, 3, 3, 0, -3, 0], [4, 0, 9, -9, 0, -4], [0, -8, 5, 6, 5, -8], [3, 6, 0, 6, 3, 0], [-5, 9, -7, -2, 0, 2], [0, -4, 2, 3, 2, -4], [9, 9, -2, -9, 0, 2], [0, 7, 7, 0, -5, -2], [-4, 0, 4, 1, 9, -3], [-15, 5, -1, 15, -4, -4], [10, 10, 0, -3, -7, 0], [-2, -3, 1, 3, 0, -3], [-17, 3, -15, 15, -3, 17], [3, -15, -15, 0, 12, 3], [6, 4, 4, 0, -16, 12], [8, 0, 7, -7, 0, -8], [0, 2, -10, 0, -10, 2], [-6, 4, 0, 2, 2, -15], [-8, 1, 15, 1, -8, 0], [1, 1, 6, -14, 13, -6], [4, 0, -4, -5, 3, -1], [-7, 0, 7, -7, -9, -14], [-3, 5, -6, 6, -5, 3], [0, 15, 0, 7, 0, 15], [11, 2, -3, 2, 11, 0], [-9, 1, -9, 0, 0, 0], [0, 4, -5, 3, -5, 4], [-15, 0, 15, 3, -1, -12], [-9, -11, 10, 9, 1, 1], [-3, -3, 7, 3, 0, -7], [5, 5, 0, -10, -10, 3], [0, 6, -8, -21, -8, 6], [-1, -18, -4, 4, 18, 1], [-12, -9, -18, 18, 9, 12], [3, -3, -16, -3, 19, 19], [0, 6, 12, -15, 12, 6], [-4, 0, 4, 8, 0, -8], [4, -7, 0, 3, 3, -12], [13, 3, -6, -13, 3, 3], [6, 17, 17, 0, -3, -14], [6, 0, -6, -12, -9, -6], [-8, -8, -3, 1, 7, 3], [-5, 6, -11, -6, 0, 6], [7, -9, 20, 13, 0, -13], [6, 6, 22, 6, -12, -22], [-3, 2, 2, 0, 11, -13], [-6, 0, 6, -3, 8, -9], [-13, 14, 0, -1, -1, 9], [4, -6, -24, -6, 4, 0], [3, -15, 5, 2, 0, -2], [7, 7, 0, -14, -14, 3], [-12, -14, -2, 12, 16, 16], [6, -19, -19, 0, 6, 13], [-12, 3, -9, 9, -3, 12], [-3, -3, 8, 3, -5, -5], [0, -9, 15, 0, 15, -9], [-24, 9, -12, 9, -24, 0], [7, -3, 12, -7, -9, -9], [-4, 0, 4, -10, -9, -14], [1, -9, 6, -6, 9, -1], [-7, -7, -9, -7, 14, 9], [-22, 17, -3, 17, -22, 0], [0, 25, -11, 12, -11, 25], [0, 0, 9, 5, -5, -9], [-5, -5, -15, -15, 20, 15], [7, 3, -3, 3, 7, 0], [-14, 13, 0, 1, 1, -3], [27, 11, -4, -27, -7, -7], [9, -4, -4, 0, 2, 2], [-3, 1, 1, 0, -4, 3], [10, -9, -3, -10, 12, 12], [-8, 6, -13, -5, 0, 5], [22, 1, 9, 1, 22, 0], [3, 3, 6, -9, 6, -6], [-21, -14, 9, 21, 5, 5], [6, -2, -2, 0, -1, 3], [9, 2, -21, 21, -2, -9], [0, 0, 24, 12, -12, -24], [0, 14, -12, 24, -12, 14], [-5, -18, 9, -18, -5, 0], [-3, 12, 0, 0, -12, 3], [16, 0, -16, -24, -3, -8], [3, -25, -25, 0, 8, 17], [9, 20, -13, -9, -7, -7], [-14, 7, 0, 7, 7, -3], [-15, 6, 20, 6, -15, 0], [0, -5, 8, -9, 8, -5], [-6, 7, 4, 6, -11, -11], [-6, 4, 4, 0, 4, -8], [1, 12, -6, -1, -6, -6], [-5, -27, -13, -8, 0, 8], [-6, 1, 0, 5, 5, 0], [15, 0, -15, 3, -6, 18], [7, -27, 6, -6, 27, -7], [0, 0, -3, -5, -3, 0], [10, 18, 4, -6, 0, 6], [0, -15, 9, 9, 0, -9], [-18, -18, -18, 8, 10, 18], [-26, 0, 26, 24, 0, -2], [-3, 3, 3, 0, 6, -9], [-10, -10, 3, -17, 27, -3], [22, -8, 0, -14, -14, -15], [0, 21, -8, 3, -8, 21], [-21, 20, 20, 0, -13, -7], [0, 11, -1, 24, -1, 11], [-15, 9, 0, 6, 6, 7], [15, 9, 9, 9, 15, 0], [3, -3, 0, 0, 0, 31], [20, 20, -12, -7, -13, 12], [-9, -10, -2, 9, 12, 12], [-2, 0, 2, -8, 27, -10], [12, -6, 14, 2, 0, -2], [16, -18, -7, -23, 0, 23], [18, -7, -12, -7, 18, 0], [0, -7, 5, 18, 5, -7], [-3, 0, 3, -7, -24, -10], [-6, 10, 7, 6, -17, -17], [12, 12, -9, 14, -26, 9], [17, -31, 0, 14, 14, 12], [18, -13, 18, -18, 13, -18], [-3, 15, 15, 0, -33, 18], [24, 28, -21, -24, -7, -7], [20, -9, -5, -25, 0, 25], [-28, 21, 0, 7, 7, -9], [-20, -20, -12, 13, 7, 12], [-4, 0, 4, 29, -9, 25], [-12, 0, 12, 0, 6, -12], [-12, 3, -10, 2, 0, -2], [-14, -6, -22, -8, 0, 8], [0, 5, -16, -9, -16, 5], [-10, -12, -5, 5, 12, 10], [-6, 0, 6, 8, -9, 2], [31, 31, -6, -15, -16, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_114_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_114_2_i_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_114_2_i_a(:prec:=6)
    chi := MakeCharacter_114_i();
    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_114_2_i_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_114_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_114_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, 162, 162, 90, 36, 9, 1]>],Snew);
    return Vf;
end function;