
// Make newform 418.2.n.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_418_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_418_n_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_418_2_n_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_418_2_n_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, 0, 1, -1, 1, 0, -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_418_n();"
function MakeCharacter_418_n()
    N := 418;
    order := 15;
    char_gens := [343, 287];
    v := [6, 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_418_n_Hecke();"
function MakeCharacter_418_n_Hecke(Kf)
    N := 418;
    order := 15;
    char_gens := [343, 287];
    char_values := [[0, 0, -1, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, -1, 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, -1, 0], [-2, 2, 0, -2, 2, -2, 0, 2], [-1, -1, 2, -1, 1, 0, -2, 1], [0, 0, -1, -2, 0, 0, -2, -1], [0, 0, 2, -1, 0, 0, 2, 2], [-3, 3, 0, 0, 0, -3, 0, 3], [-6, -1, 7, -6, 6, 0, -7, 6], [0, 2, 0, 3, -2, -2, 2, -2], [1, 0, 0, 0, 0, 1, 0, 0], [-2, 0, 0, 0, 2, -2, 0, 1], [-3, 0, 3, 0, 0, 0, 5, 3], [0, 0, 5, -1, 0, 0, -1, 5], [-5, -3, 0, -5, 5, 3, -8, 5], [6, -6, -6, 6, -6, -1, 0, -6], [1, -1, 0, 0, -1, 1, 0, -1], [-2, 10, 0, 0, 0, -2, 0, 2], [6, 0, -1, 6, -6, 6, 6, -6], [0, 3, 0, 0, -9, 0, 0, 3], [-1, -7, 0, 0, -7, -1, 0, 0], [-2, 7, -5, -2, 2, 0, 5, 2], [-2, 0, -8, -2, 2, -2, -2, 2], [0, 6, -6, 0, 0, -6, 6, 0], [-9, 0, 5, -9, 0, 0, -5, 5], [-5, -6, 0, 0, -6, -5, 0, 0], [0, 0, 10, 0, 0, 10, 0, 0], [-2, 6, 0, 0, 0, -2, 0, 2], [0, 0, 4, -8, 0, 0, -8, 4], [5, 0, 0, 3, 0, 0, 5, 0], [5, -5, -5, 5, -5, 16, 0, -5], [13, 0, 0, 3, 0, 0, 13, 0], [0, -9, 0, 0, 5, 0, 0, -9], [-4, 4, 4, -4, 4, -3, 0, 4], [0, -13, 0, 0, -3, 0, 0, -13], [12, 0, 0, 0, -12, 12, 0, -3], [-6, 5, 1, -6, 6, 0, -1, 6], [-2, 0, 0, 9, 0, 0, -2, 0], [-8, 7, 0, -8, 8, -7, -1, 8], [8, 0, -8, 0, 0, 0, 8, -8], [0, 17, 0, 0, 0, 0, 0, 0], [-10, -8, 0, -10, 10, 8, -18, 10], [-6, 0, 0, 2, 0, 0, -6, 0], [0, -2, 0, 0, 12, 0, 0, -2], [0, 0, 10, -6, 0, 0, -6, 10], [1, -12, 11, 1, -1, 0, -11, -1], [20, 0, -9, 9, 0, 0, 0, -9], [-16, -3, 0, 0, -3, -16, 0, 0], [0, -17, 5, 0, 0, 5, -17, 0], [-18, 14, 0, -18, 18, -14, -4, 18], [0, 0, -7, 0, 0, 0, 0, -7], [-6, 0, 6, 0, 0, 0, 9, 6], [0, 15, -3, 0, 0, -3, 15, 0], [-7, 0, 0, -21, 0, 0, -7, 0], [-6, 7, 0, 0, 7, -6, 0, 0], [-3, -16, 0, 0, 0, -3, 0, 3], [0, 0, 0, 0, 0, 0, 0, 14], [-11, 11, 11, -11, 11, -1, 0, 11], [-11, 7, 4, -11, 11, 0, -4, 11], [1, 0, 1, 1, -1, 1, 1, -1], [12, 0, -12, 0, 0, 0, 7, -12], [0, -9, 0, 0, 2, 0, 0, -9], [-17, 19, 0, -17, 17, -19, 2, 17], [0, 0, -1, 12, 0, 0, 12, -1], [-15, 15, 15, -15, 15, -16, 0, 15], [-9, 0, 0, -10, 0, 0, -9, 0], [0, 8, 0, 0, -10, 0, 0, 8], [6, -11, 0, 0, 0, 6, 0, -6], [4, 0, -13, 13, 0, 0, 0, -13], [0, 0, 23, 0, 0, 0, 0, 0], [4, -2, -2, 4, -4, 0, 2, -4], [-6, 0, 0, 21, 0, 0, -6, 0], [4, 0, 0, 0, 0, 0, 0, 0], [10, 0, 7, 10, -10, 10, 10, -10], [23, 0, 0, 0, -23, 23, 0, -13], [-3, 0, 20, -20, 0, 0, 0, 20], [12, 0, -5, 12, 0, 0, 5, -5], [0, -22, -2, 0, 0, -2, -22, 0], [11, 0, 0, 0, -11, 11, 0, -7], [-8, 8, 8, -8, 8, -14, 0, 8], [14, -8, -6, 14, -14, 0, 6, -14], [-9, -8, 0, 0, 0, -9, 0, 9], [-25, 0, 14, -14, 0, 0, 0, 14], [-5, 9, 0, -5, 5, -9, 4, 5], [-4, -15, 0, 0, 0, -4, 0, 4], [-7, 0, 0, 0, 7, -7, 0, -7], [24, -24, -24, 24, -24, 3, 0, -24], [-25, 25, 0, 0, 2, -25, 0, 2], [-5, 0, 5, 0, 0, 0, -3, 5], [-1, 0, -15, -1, 0, 0, 15, -15], [-14, 14, 14, -14, 14, -19, 0, 14], [7, 0, -6, 6, 0, 0, 0, -6], [6, 0, 14, 6, 0, 0, -14, 14], [0, 24, 0, 0, 17, 0, 0, 24], [-8, 0, 0, 19, 0, 0, -8, 0], [18, 0, -18, 18, -18, 18, 18, -18], [31, 0, -8, 31, -31, 31, 31, -31], [16, -16, 0, 0, 5, 16, 0, 5], [-1, 1, 0, 0, -15, -1, 0, -15], [0, 0, -3, 16, 0, 0, 16, -3], [0, -19, 0, 0, -8, 0, 0, -19], [21, -5, 0, 0, 0, 21, 0, -21], [11, -11, 0, 0, -9, 11, 0, -9], [22, 0, 0, 0, -22, 22, 0, -13], [18, 0, -18, 0, 0, 0, 16, -18], [30, 0, 0, 4, 0, 0, 30, 0], [-7, 0, -13, 13, 0, 0, 0, -13], [-29, 0, 20, -29, 0, 0, -20, 20], [10, 0, -36, 10, -10, 10, 10, -10], [30, 4, 0, 0, 4, 30, 0, 0], [0, -21, 0, 0, -1, 0, 0, -21], [0, 0, 2, -14, 0, 0, -14, 2], [-38, 0, 38, 0, 0, 0, -19, 38], [1, -16, 0, 1, -1, 16, -15, -1], [26, -18, 0, 0, -18, 26, 0, 0], [18, 0, 0, 27, 0, 0, 18, 0], [-14, 0, 0, 0, 14, -14, 0, 27], [9, 2, 0, 9, -9, -2, 11, -9], [0, -22, 22, 0, 0, 0, -22, 0], [9, 0, -9, 0, 0, 0, 4, -9], [0, 0, 34, -18, 0, 0, -18, 34], [-35, 7, 0, 0, 7, -35, 0, 0], [1, 3, 0, 0, 3, 1, 0, 0], [16, 0, -16, 0, 0, 0, -14, -16], [-4, 0, -10, -4, 0, 0, 10, -10], [11, 0, -6, 6, 0, 0, 0, -6], [4, 0, -4, 0, 0, 0, -17, -4], [-5, -1, 0, -5, 5, 1, -6, 5], [0, 0, 0, 0, 49, 0, 0, 0], [-11, 0, 8, -11, 11, -11, -11, 11], [-9, 9, 9, -9, 9, 17, 0, 9], [0, 0, -13, 20, 0, 0, 20, -13], [0, -23, 24, 0, 0, 24, -23, 0], [37, -16, -21, 37, -37, 0, 21, -37], [9, -9, 0, 0, 19, 9, 0, 19], [0, -31, 16, 0, 0, 16, -31, 0], [2, 0, -2, 0, 0, 0, -17, -2], [-16, 15, 0, 0, 15, -16, 0, 0], [24, -24, 0, 0, -9, 24, 0, -9], [23, 0, -33, 23, 0, 0, 33, -33], [-12, 0, 12, -12, 0, 0, -12, 12], [-8, 0, 1, -8, 0, 0, -1, 1], [18, -18, 0, 0, -2, 18, 0, -2], [-24, 0, 0, 0, 24, -24, 0, 30], [-21, 20, 0, 0, 0, -21, 0, 21], [-13, 0, 0, 0, 0, -13, 0, 13], [0, 0, 21, 6, 0, 0, 6, 21], [-37, 16, 0, -37, 37, -16, -21, 37], [25, -6, -19, 25, -25, 0, 19, -25], [30, -30, -30, 30, -30, 8, 0, -30], [5, 32, 0, 0, 32, 5, 0, 0], [-49, 0, 54, -49, 0, 0, -54, 54], [42, -22, 0, 42, -42, 22, 20, -42], [30, 0, -12, 12, 0, 0, 0, -12], [7, -7, 0, 0, -1, 7, 0, -1], [18, 0, 0, 0, -18, 18, 0, -35], [7, 12, -19, 7, -7, 0, 19, -7], [15, 0, -15, 0, 0, 0, 35, -15], [-30, 0, -10, -30, 0, 0, 10, -10], [28, 2, -30, 28, -28, 0, 30, -28], [-15, -13, 0, 0, 0, -15, 0, 15], [0, 4, 0, 0, 8, 0, 0, 4], [7, -7, -7, 7, -7, 26, 0, -7], [-22, 0, -9, -22, 22, -22, -22, 22], [13, -13, -13, 13, -13, -3, 0, -13], [-17, 38, 0, -17, 17, -38, 21, 17], [-31, 0, 31, 0, 0, 0, -29, 31], [-24, 0, -5, -24, 24, -24, -24, 24], [-15, 15, 15, -15, 15, -22, 0, 15], [3, 0, 0, 0, -3, 3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_418_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_418_2_n_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_418_2_n_c(:prec:=8)
    chi := MakeCharacter_418_n();
    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_418_2_n_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_418_2_n_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_418_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, -128, 0, 32, -16, 8, 0, -2, 1]>],Snew);
    return Vf;
end function;