
// Make newform 414.2.e.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_414_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_414_e_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_414_2_e_d();".
// 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_414_2_e_d();".
// 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 := [9, -21, 64, -1, 64, -19, 34, -4, 7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-98232, -130790, -30472, -47450, -51907, -25351, -10326, -4352, -2394, -658], [-9081, -194145, -41986, -65255, -20431, -34543, -4688, -4321, -537, -999], [-29709, 15165, -44484, 51145, -36059, 19883, -7882, 4381, -1348, 339], [245043, -110550, 110748, -119555, 89773, -49501, 16239, -10907, 3356, -1348], [59532, 176015, 188992, 165845, 17167, 84646, 3021, 17417, -1626, 2533], [-139464, 342765, -180434, 369510, -159864, 133958, -34722, 26226, -6203, 2694], [-54156, 166780, 72109, 175205, -42836, 104352, -8823, 21759, -2962, 3301], [11514, -51320, -30640, -51488, -3072, -27923, -998, -5659, 248, -840], [-134607, 258905, -54152, 343585, -179927, 193324, -39341, 41363, -8994, 6042]];
    Rf_basisdens := [1, 72795, 72795, 24265, 72795, 72795, 72795, 72795, 14559, 72795];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_414_e();"
function MakeCharacter_414_e()
    N := 414;
    order := 3;
    char_gens := [47, 235];
    v := [1, 3];
    // 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_414_e_Hecke();"
function MakeCharacter_414_e_Hecke(Kf)
    N := 414;
    order := 3;
    char_gens := [47, 235];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 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 := [[1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, 0, 0, 0, 1, 1, 0], [-3, -1, -2, 1, -1, 2, -1, -3, 2, 1], [0, 2, 0, 0, 1, 0, -1, -1, 0, -1], [0, 0, 2, -2, 2, 1, 2, 0, 0, -1], [-1, 0, 1, 1, 1, 1, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, -1, -1, -2, 1, 0, 0, -3], [0, 2, 2, -2, 1, 0, -1, -1, 0, -1], [-4, -2, -2, -1, 0, 1, -2, 0, 2, 1], [0, -1, 0, 0, 1, -3, 0, 6, -2, 1], [4, 0, -1, 1, 0, 2, -1, 4, 2, 2], [-3, 0, 2, 0, 4, 0, 0, -3, -4, 4], [-3, 0, -1, 3, -1, -1, -1, 0, 0, 1], [0, -3, 0, 0, -2, 1, -1, 0, -2, 4], [1, 5, 1, -1, 0, -2, 1, 1, 3, -2], [0, -1, 1, -1, 0, -1, 0, -1, -1, 1], [6, 1, -1, 3, -2, -2, -1, 0, -1, 1], [2, -1, -1, 2, 0, -1, -1, 0, 1, 2], [-2, 6, 5, -2, 4, -4, 2, -2, -2, 0], [-6, 1, -1, 1, -1, 2, -1, -6, 4, 1], [9, 0, -3, 1, -3, 1, -3, 0, 0, -1], [4, -1, 0, 1, 1, 2, -1, 4, 0, 3], [-6, -5, -5, 2, -2, 4, -2, -6, 1, 2], [0, -6, -4, 4, -2, -2, -1, -7, -5, 7], [-6, 0, 3, -2, 3, -1, 3, 0, 0, 1], [2, 1, 4, -5, 3, 2, 4, 0, -1, -3], [0, -2, -2, 2, 1, -4, -2, 3, -5, 4], [2, 0, 1, -2, 1, -2, 1, 0, 0, 2], [0, 5, -1, 1, 1, 3, 1, 3, 5, -6], [-6, -3, 1, 0, 2, 0, 0, -6, -5, 2], [0, -2, -1, 1, 1, -4, 2, 5, -1, 0], [0, -4, -2, 2, -4, 4, 0, -3, 0, 4], [-2, -3, -3, 2, -3, 4, -2, -2, 4, 1], [0, 0, -4, 4, 0, 0, -4, -4, -4, 4], [2, -1, -4, 2, -3, 7, -4, 0, 1, -6], [0, -3, -3, 3, -1, -1, -4, -6, -6, 7], [3, -4, -2, 0, 0, 0, 0, 3, -4, 0], [6, 0, 1, 1, 1, 1, 1, 0, 0, -1], [2, 3, 4, -2, 1, -7, 4, 0, -3, 4], [6, 6, 0, 0, 1, 0, 0, 6, 5, 1], [0, -4, -1, 1, 1, -6, 1, 8, -4, 3], [0, 1, 4, -5, 3, -3, 4, 0, -1, 2], [-1, 0, -2, -1, -2, 4, -2, 0, 0, -4], [0, -6, 0, 0, -2, -2, 1, 2, -3, 5], [1, -4, -4, 1, 3, 2, -1, 1, -5, 5], [-3, 7, 3, -1, 4, -2, 1, -3, 1, 2], [0, 4, -1, 1, 3, -2, 0, 2, 1, -4], [3, 3, 4, 0, 1, 2, 4, 0, -3, -5], [0, -5, 1, -1, 2, -9, 5, 6, -2, 0], [7, 3, 8, -5, 0, -10, 5, 7, -7, -10], [9, 1, 5, -7, 4, 1, 5, 0, -1, -2], [0, -8, -6, 6, -3, -2, 0, -3, -5, 8], [0, 8, 1, -2, -4, -4, 2, 0, 8, -8], [-9, 0, -3, -3, -3, -1, -3, 0, 0, 1], [8, 3, -1, 0, -4, -2, -1, 0, -3, -1], [-5, -9, -3, 3, -8, 6, -3, -5, 5, -2], [-6, 8, 9, -1, 3, -2, 1, -6, 3, 1], [0, 1, -3, 3, -1, 3, 2, -4, 4, -3], [0, -2, 3, -3, -1, 0, 1, 3, 0, 1], [-7, 4, 10, -3, 6, -4, 10, 0, -4, 0], [0, 3, -4, 4, 0, 3, -6, -6, -3, 3], [-2, 7, 4, -4, 1, -8, 4, -2, -2, -7], [-3, 5, -7, 1, -5, 2, -1, -3, 12, -3], [-2, 2, 6, -3, 4, -6, 3, -2, -8, -2], [0, -8, -5, 5, -4, 0, -5, -4, -9, 13], [0, 5, 6, -6, 4, -3, -1, -9, 0, -4], [-11, 0, -2, 0, 5, 0, 0, -11, -5, 5], [-9, -6, -1, 2, 1, 4, -2, -9, -3, 5], [3, -6, 1, -5, 7, 0, 1, 0, 6, 6], [19, -3, 4, -1, 0, -2, 1, 19, -5, -2], [0, 0, 8, -8, 0, 0, -2, 5, -2, 2], [-7, 2, 4, 1, 2, -8, 4, 0, -2, 6], [0, 7, -2, 2, 2, 3, 5, -6, 10, -12], [-9, -4, -6, 3, -2, 6, -3, -9, 4, 4], [8, 0, -2, 3, -2, -4, -2, 0, 0, 4], [0, 1, -2, 2, -2, 5, -6, 6, -3, 5], [0, -2, -10, 10, -1, 0, -3, 14, -4, 5], [0, -4, -2, 2, 0, -4, 3, 12, -1, 1], [-8, 5, 14, -2, 7, -4, 2, -8, -6, 3], [-3, 1, 2, -8, 1, -4, 2, 0, -1, 3], [17, 1, 3, 4, 2, -1, 3, 0, -1, 0], [1, -10, -4, 5, 1, 10, -5, 1, -1, 11], [6, -12, -3, 4, -1, 8, -4, 6, -3, 7], [-3, 5, 2, 1, -3, -3, 2, 0, -5, -2], [-5, 10, 6, -4, 1, -8, 4, -5, 1, -7], [3, 4, 12, -3, 2, -6, 3, 3, -4, -4], [0, -8, -6, 6, 0, -8, -2, 8, -10, 10], [-3, -2, -3, 10, -1, 3, -3, 0, 2, -1], [-12, 13, 13, -6, 12, -12, 6, -12, -11, 0], [8, 5, 0, 1, -5, -1, 0, 0, -5, -4], [0, 2, -9, 9, -3, 8, -8, -6, -3, 6], [0, 8, 7, -7, 2, 4, 0, -4, 6, -8], [-6, -5, -5, 8, 0, 10, -5, 0, 5, -5], [0, -2, -6, 6, -5, 8, -5, -3, -2, 7], [6, -5, -7, -12, -2, 2, -7, 0, 5, 3], [-13, -4, -8, 5, -4, 7, -8, 0, 4, -3], [5, 3, 10, -9, 7, -11, 10, 0, -3, 8], [16, -13, -8, 2, -5, 4, -2, 16, -4, -1], [-6, 1, -3, 2, -4, -5, -3, 0, -1, 4], [0, 2, 1, -1, -2, 6, -6, 3, -2, 4], [6, -9, 1, 0, -6, 0, 0, 6, -3, -6], [0, -2, 6, -6, 4, -10, 4, 23, -2, -2], [-16, -7, -6, 3, 1, 17, -6, 0, 7, -10], [-3, -12, -6, 5, 0, 10, -5, -3, -2, 10], [-9, -2, 1, -3, 3, 8, 1, 0, 2, -6], [0, 1, -9, 9, -4, 9, -2, 3, 3, 1], [-17, 8, 8, -2, 1, -4, 2, -17, 3, -3], [0, -9, -13, 13, -7, 5, -7, -16, -9, 16], [5, 2, 8, -10, 6, 3, 8, 0, -2, -5], [0, -14, 0, 0, 1, -16, 4, 3, -11, 10], [1, 5, -3, 0, -7, 0, 0, 1, 12, -7], [-1, 1, 6, 1, 5, 7, 6, 0, -1, -8], [0, -16, -7, 7, -14, 14, -7, 0, 12, 0], [0, 11, -1, 1, 3, 5, -2, -16, 6, -9], [18, 1, -7, -3, -8, 2, -7, 0, -1, -3], [0, 0, 3, -3, 5, -10, -5, 0, -10, 5], [15, -2, -2, 1, -7, 2, -1, 15, 7, -5], [0, 14, 1, -1, 4, 6, -3, 2, 7, -11], [-11, -2, 6, -1, 14, -2, 1, -11, -18, 12], [6, -11, -3, 4, 3, 8, -4, 6, -6, 11], [-18, -2, 5, 0, 7, 7, 5, 0, 2, -5], [4, 5, 7, -4, 11, -8, 4, 4, -14, 3], [-6, 6, 8, 8, 2, -3, 8, 0, -6, -3], [10, 3, -6, 0, -3, 0, 0, 10, 6, -3], [-15, 4, 4, 7, 0, 0, 4, 0, -4, -4], [-23, -16, -9, 7, -2, 14, -7, -23, 0, 12], [0, -1, -6, 6, 2, -5, 7, -1, 4, -6], [2, 6, 2, -1, -4, -14, 2, 0, -6, 8], [0, 3, 6, -6, 2, -1, 10, -12, 11, -13], [0, 3, -1, 1, 1, 1, 4, -19, 6, -7], [-10, 4, -6, -1, -10, -8, -6, 0, -4, 4], [0, -2, 3, -3, 2, -6, 6, 12, 2, -4], [0, -14, 2, -2, 2, -18, 3, 11, -13, 11], [0, 2, 9, -7, 7, -12, 9, 0, -2, 10], [0, 7, -4, 4, -4, 15, -1, -22, 10, -6], [0, 7, 0, 0, 5, -3, -1, -6, 1, -6], [-21, 3, -1, -5, -4, -1, -1, 0, -3, -2], [-7, -5, 0, -11, 5, 1, 0, 0, 5, 4], [-12, -5, -10, 8, -1, 16, -8, -12, 12, 15], [0, 6, 10, -10, -1, 8, 13, -10, 20, -19], [-15, -8, -11, 7, -3, 12, -11, 0, 8, -4], [11, 3, 8, 3, 5, 6, 8, 0, -3, -9], [-6, -10, -16, 5, -11, 10, -5, -6, 11, -1], [16, -9, -10, 5, -8, 10, -5, 16, 9, 2], [15, 7, 11, -4, 2, -8, 4, 15, -3, -6], [0, -8, -3, 3, 3, -14, 7, 2, -4, 1], [-27, -3, -3, 1, 0, 3, -3, 0, 3, 0], [0, -1, 11, -11, 2, -5, 7, -7, 4, -6], [6, -1, 6, -2, 7, -5, 6, 0, 1, 6], [20, 3, 5, 0, 2, -12, 5, 0, -3, 9], [0, 13, 10, -10, 11, -9, 2, 3, 4, -15], [-14, -25, -15, 7, -9, 14, -7, -14, -2, 5], [-6, 27, 16, -10, 17, -20, 10, -6, -10, -3], [-19, -3, -3, 4, 0, 13, -3, 0, 3, -10], [-21, -2, -4, 2, -7, 4, -2, -21, 9, -3], [38, 0, 1, -3, 1, -3, 1, 0, 0, 3], [0, 11, 5, -5, 4, 3, 4, 6, 11, -15], [9, -12, -16, 8, -9, 16, -8, 9, 13, 7], [-9, -4, -9, 11, -5, -8, -9, 0, 4, 12], [0, -1, -2, 2, -4, 7, -2, -4, 1, 3], [9, 1, -5, 4, -6, -10, -5, 0, -1, 9], [0, 0, -7, 7, 4, -8, -6, 6, -10, 6], [-21, 18, 16, -7, 16, -14, 7, -21, -12, 2], [-7, -7, -1, 0, 6, 17, -1, 0, 7, -10], [-2, 16, 12, -7, 14, -14, 7, -2, -12, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_414_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_414_2_e_d();".
// 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_414_2_e_d(:prec:=10)
    chi := MakeCharacter_414_e();
    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_414_2_e_d();".
// 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_414_2_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_414_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, -225, 580, -341, 334, 1, 124, -2, 13, 1, 1]>],Snew);
    return Vf;
end function;