
// Make newform 245.2.l.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_245_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_245_l_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_245_2_l_b();".
// 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_245_2_l_b();".
// 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, -1, 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_245_l();"
function MakeCharacter_245_l()
    N := 245;
    order := 12;
    char_gens := [197, 101];
    v := [9, 10];
    // 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_245_l_Hecke();"
function MakeCharacter_245_l_Hecke(Kf)
    N := 245;
    order := 12;
    char_gens := [197, 101];
    char_values := [[0, 0, 0, 1], [0, 0, 1, 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 := [[1, 0, -1, -1], [1, 1, 0, -1], [0, -1, -2, 1], [0, 0, 0, 0], [1, -1, -1, 2], [2, 0, 0, 2], [-2, 2, 0, -2], [0, 1, 1, 1], [3, 4, 1, -1], [0, 0, 0, -3], [2, -4, 2, 0], [-4, 2, 2, 2], [2, 0, -4, 3], [-4, -5, 5, 4], [-4, 1, 5, -5], [5, -5, -5, 0], [3, -3, -3, 6], [4, 5, -2, -5], [-1, -1, 6, -5], [3, 2, 0, -1], [-4, -4, 8, -4], [-2, -5, 1, 5], [1, -3, -3, 1], [0, -5, -8, -5], [-5, -8, 8, 5], [1, 10, 1, 0], [-1, 3, -2, -2], [9, -6, -3, -3], [-7, -2, -7, 0], [-5, 2, -2, 5], [-1, -4, -4, -1], [4, -12, -2, 12], [-4, -4, -6, 10], [-3, -10, 0, 5], [-6, -6, 3, 6], [0, 8, 0, -16], [-1, 1, 13, 12], [-8, -1, 7, -7], [6, 13, -13, -6], [-18, 8, 10, 10], [8, 6, 8, 0], [3, 0, -6, -4], [0, -1, -15, -1], [-5, 5, 3, -2], [-5, 14, 14, -5], [18, 4, -18, -8], [5, 6, 0, -3], [-13, 14, 14, -13], [11, -11, -3, 8], [0, -6, 8, -6], [-2, 2, 4, -4], [6, 0, -12, -8], [-12, 4, -12, 0], [-8, 0, 16, -8], [0, -2, -2, 2], [-10, 10, 7, -3], [-9, 8, 9, -16], [2, 23, -1, -23], [12, 12, -6, -6], [6, 8, 0, -4], [15, 15, -3, -12], [-2, -12, -12, -2], [6, 9, -9, -6], [-1, -17, -1, 0], [-6, -6, 12, 12], [2, -6, 4, 4], [0, -8, -12, -8], [11, 4, 4, 11], [9, 9, -18, 9], [-8, -2, 0, 1], [-1, -1, -7, 8], [28, -10, -14, 10], [0, 0, -1, -1], [10, 8, -2, 2], [-5, 0, 10, 11], [-1, 22, -21, -21], [-4, -2, -4, 0], [10, 11, 1, -1], [0, 0, -11, 0], [7, -8, -7, 16], [-10, 16, 0, -8], [-26, -10, 0, 5], [-1, 3, 1, -6], [19, 4, 4, 19], [0, 10, 14, 10], [-7, 0, 7, -7], [-11, 0, 22, -14], [-18, 2, 16, 16], [-6, 0, 12, 16], [13, 13, -13, -13], [-19, -16, 3, -3], [-20, -4, 20, 8], [-9, -9, 17, -8], [10, -32, 0, 16], [-26, -11, 13, 11], [18, 1, 1, 18], [0, -19, 6, -19], [8, -10, 8, 0], [12, -27, 15, 15], [0, 16, -9, 16], [-12, 13, 13, -12], [-1, -1, 17, -16], [-16, -16, 17, -1], [-8, 22, 4, -22], [-1, -11, 1, 22], [1, -1, -15, -14], [17, 24, -24, -17], [-6, 13, -7, -7], [-16, -10, -16, 0], [18, 0, -36, 10], [-9, -3, 6, -6], [12, -12, 4, 16], [20, -38, -38, 20], [-5, 3, 5, -6], [11, 18, 0, -9], [-2, 5, 2, -10], [-19, -4, -4, -19], [-22, 22, 3, -19], [-12, -2, 10, -10], [-5, 0, 10, -19], [18, -17, 18, 0], [-12, -12, 12, 12], [8, -8, -16, 16], [7, -7, 8, 15], [46, -11, -23, 11], [-22, -2, 0, 1], [8, -15, -4, 15], [0, 9, -23, 9], [6, -3, 3, -6], [6, -1, -5, -5], [-9, 7, -9, 0], [-13, -5, 5, 13], [0, -6, -4, -6], [12, -2, -2, 12], [-24, 8, 12, -8], [16, 36, 0, -18], [-8, -8, -6, 14], [-21, 21, 16, -5], [-28, -4, 4, 28], [6, 15, 6, 0], [-4, 0, 8, -36], [0, -10, 32, -10], [-27, 27, -6, -33], [-27, 17, 17, -27], [18, 2, -18, -4], [-19, -14, 0, 7], [13, -14, -14, 13], [3, -3, 11, 14], [0, -4, 28, -4], [7, -27, -34, 34], [-10, 25, -15, -15], [8, 0, -16, -39], [19, 8, -8, -19], [11, 23, 12, -12], [6, 6, -19, 13], [15, 26, 0, -13], [-6, 51, 3, -51], [0, -10, 19, -10], [-28, -8, 8, 28], [-20, 6, -20, 0], [-19, -6, 25, 25], [-16, 16, -16, 16], [-11, 7, 7, -11], [-48, -8, 24, 8], [25, 25, -7, -18], [0, 0, -29, 29], [-4, 16, 4, -32], [27, -27, -21, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_245_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_245_2_l_b();".
// 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_245_2_l_b(:prec:=4)
    chi := MakeCharacter_245_l();
    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_245_2_l_b();".
// 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_245_2_l_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_245_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -4, 5, -2, 1]>],Snew);
    return Vf;
end function;