
// Make newform 414.2.e.c 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_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_414_2_e_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 := [243, -162, 162, -99, 66, -45, 22, -11, 6, -2, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [648, 1188, 198, 57, 135, -32, -47, -3, 25, 1], [-810, -351, -720, 606, -153, 124, 1, 51, -26, 4], [-2997, 891, -810, 528, -339, 218, -121, 42, -37, 2], [-810, -162, 729, -318, 372, -37, 92, -75, 23, -10], [-2511, 3240, -1350, 1068, -573, 425, -139, 51, -19, 11], [-891, -162, -135, -321, 357, -5, 64, -15, 25, -14], [162, 891, -189, 204, -132, 83, -58, 33, -10, 11], [3321, -3564, 2079, -1425, 1011, -451, 281, -111, 68, -37]];
    Rf_basisdens := [1, 1, 1701, 1701, 1701, 1701, 1701, 1701, 567, 1701];
    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, 1, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, -1, 0, 0, -1], [0, 0, 0, -1, -2, -1, 0, 1, 0, 0], [2, 0, 1, 1, 1, 0, -1, 1, -1, 0], [1, 2, 1, 0, 0, 0, -1, 0, -1, -2], [0, 0, 0, 1, -1, 1, 0, -1, -1, -1], [0, 3, 1, 2, 1, 1, -2, -1, -1, 1], [-2, 1, -3, 0, -3, -2, -1, 3, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [4, 2, -1, 0, 3, 2, -1, 0, -1, -2], [0, -2, -2, -5, -7, -3, -2, 3, 1, -1], [1, 1, 0, -1, 0, 3, -2, 0, 1, -1], [-1, 2, 1, -1, -6, -3, 2, 2, -2, 0], [1, 3, 2, 3, -3, 2, -4, 3, -4, -3], [3, 2, 0, 0, 1, 2, -2, 0, -2, -2], [-6, 1, -1, 0, -1, 0, -1, 1, 0, 0], [-1, 2, 1, 6, 2, 4, 2, -5, -1, 1], [0, -3, -2, 0, 5, -3, 5, 0, 5, 3], [-2, 1, -1, 4, 5, 3, 1, -5, 4, 5], [2, -3, 0, -4, 0, 4, 0, 0, 3, -3], [-3, -4, 0, -1, 0, -3, 4, 0, 0, 0], [3, 0, 1, 0, 3, -1, 0, 0, 0, 0], [3, 0, -2, -2, 4, 1, 1, -2, 1, 0], [0, -1, 3, 2, 3, 2, -1, -3, 2, -2], [-8, 2, -3, -2, -7, 2, 1, -2, 1, -2], [3, -4, -3, 1, 8, -2, 5, 1, 5, 4], [7, -3, 4, -1, -3, 2, -3, 5, -5, -8], [-8, -1, 2, -1, 2, 0, 3, -2, -2, 2], [4, -1, 1, -3, 1, 1, 3, -1, -2, 2], [-3, 3, 0, 4, 3, 1, 3, -4, 0, 3], [-10, 0, 3, 3, 3, -1, 1, -3, -1, 1], [2, -3, -1, -1, -7, 2, -3, 0, 5, 2], [-5, -4, -5, -3, 0, 0, 5, -3, 5, 4], [1, 3, 4, 3, 5, 0, 3, 1, -2, 1], [0, 2, 2, 4, 5, 2, 2, -2, 0, 2], [-9, -5, -1, -3, -6, -2, 3, -3, 3, 5], [4, 0, 4, 0, 0, 0, 0, 4, -4, -4], [-5, 8, -1, 6, -1, -4, -3, 1, -5, 5], [-9, 2, -7, 0, 2, -2, 2, -7, 6, 8], [-1, 0, -2, -4, 1, 0, 2, -4, 2, 0], [-3, -7, -1, -2, -1, -2, 3, 1, 4, -4], [1, 6, 9, 10, 9, 0, -1, -9, -5, 5], [10, 1, 8, 2, 4, -2, -6, 2, -6, -1], [-4, 2, -2, -6, -8, -8, 2, 4, -3, -1], [2, -2, -2, 1, -2, -4, 1, 2, 1, -1], [10, 0, 3, -3, 3, 8, -2, -3, 2, -2], [3, -3, 0, 3, 2, 6, -3, -3, -3, -6], [5, 0, -1, 1, -1, 7, -6, 1, -6, 0], [6, 1, 4, 6, -1, 3, -7, 6, -7, -1], [3, 1, 4, -1, 4, -2, 1, 5, -6, -5], [6, 9, 6, 10, 6, 5, -9, -6, 0, 0], [3, 0, 3, 1, -4, 1, 0, 2, -9, -9], [-5, -1, -3, 2, -8, 6, -3, 2, -3, 1], [-1, 2, 11, 4, 11, 7, 0, -11, -2, 2], [5, -3, 2, -6, -5, -3, -3, 8, 0, -3], [-3, 4, 1, -2, -9, 5, -6, -2, -6, -4], [-22, 1, -1, 0, -1, 0, -1, 1, 0, 0], [-1, -13, -7, -13, -7, -2, 8, 7, 5, -5], [-4, 7, -2, -2, -3, 1, 1, -2, 1, -7], [-5, -4, -2, -3, 1, -4, 6, -3, 6, 4], [-3, 2, -1, 2, 0, 0, 2, -3, -4, -2], [0, -2, -2, 2, 1, 4, -2, -4, 11, 9], [-4, -6, 3, 3, 3, -8, 8, -3, -2, 2], [-8, 7, -1, 5, 2, -2, 7, -6, -4, 3], [-2, 5, 4, 1, -5, -1, -3, 1, -3, -5], [-2, 4, -2, 4, -3, 3, -1, 4, -1, -4], [0, -1, 1, 1, 4, -5, 4, 1, 4, 1], [-1, 1, 0, 2, 16, 1, 1, -2, 3, 4], [7, -6, 1, 6, 13, 12, -6, -5, 9, 3], [9, -7, 4, 6, 7, -2, -2, 6, -2, 7], [8, 5, 5, 0, 4, -1, -4, 0, -4, -5], [6, 12, 4, 6, 4, -1, -1, -4, -11, 11], [-7, -1, 3, 6, -10, 0, -3, 6, -3, 1], [2, -4, -2, -8, 1, -4, -4, 6, 2, -2], [10, -2, -11, -7, -11, -4, 0, 11, 2, -2], [-5, 2, -3, -8, -18, -10, 2, 5, -5, -3], [7, -7, 5, -5, 5, -3, -2, -5, -2, 7], [-7, -6, -9, -7, -9, -8, 6, 9, 0, 0], [-12, 7, -5, 1, -4, -6, 7, -6, 2, 9], [0, 8, 8, 9, 12, 1, 8, -1, -9, -1], [-3, -5, -8, -11, -22, -6, -5, 3, 7, 2], [-6, 7, 6, 5, -9, -3, -3, 5, -3, -7], [-2, -3, 2, 4, 2, -3, 1, -2, 2, -2], [22, -2, 10, 5, 10, 7, -2, -10, 4, -4], [-3, 2, -5, 3, 1, 1, 4, 3, 4, -2], [1, -1, -1, 6, 3, -1, 2, 6, 2, 1], [-11, 10, 0, 1, 0, 4, -5, 0, -5, 5], [-11, 1, 0, -4, -13, 2, -2, -4, -2, -1], [-3, 3, -5, 5, 1, 1, 4, 5, 4, -3], [-6, -2, -8, -4, -12, -2, -2, -4, 2, 0], [5, -11, -8, -9, -8, -8, 9, 8, 2, -2], [-7, -4, 3, 1, -10, 0, -3, 1, -3, 4], [-8, -12, 2, -1, 2, -4, 7, -2, 5, -5], [-3, 3, 0, 7, 18, 4, 3, -7, 2, 5], [-8, 4, -4, -5, 0, -9, 4, 1, 8, 12], [11, 1, -1, -7, -1, 8, -2, 1, 1, -1], [-6, 6, 0, 3, -11, -3, 6, -3, -9, -3], [-5, 1, 1, 5, 1, -6, 2, -1, -3, 3], [-2, 1, 10, 11, 10, 3, -4, -10, 3, -3], [1, -6, 0, -1, 0, 4, -3, 0, 9, -9], [4, 1, -2, -3, 11, -5, 7, -3, 7, -1], [-3, 4, 2, 7, 2, -1, -4, -2, 0, 0], [-2, 4, 2, 3, 11, -1, 4, -1, -12, -8], [-1, -4, 3, 3, -2, -2, -1, 3, -1, 4], [-4, 4, 0, 0, -5, -4, 4, 0, 6, 10], [22, -6, 6, -5, 6, 12, -1, -6, 7, -7], [5, 1, 3, -5, 3, -1, -2, -5, -2, -1], [-6, -4, -10, -6, -10, -5, 1, 10, 3, -3], [4, -1, 3, -2, 3, -1, -1, 5, -8, -9], [-9, -1, -6, 10, -11, 8, -2, 10, -2, 1], [16, -7, 9, 7, -6, 14, -7, 2, 5, -2], [3, 5, -11, -4, -11, 5, -12, 11, 7, -7], [-5, -3, -8, -8, -1, -5, -3, 0, 2, -1], [10, 9, -7, 7, 11, 6, 1, 7, 1, -9], [-11, -8, -4, 1, -4, -16, 11, 4, -3, 3], [-17, -1, 4, -7, -19, -2, -2, -7, -2, 1], [5, 0, 5, -2, 4, -2, 0, 7, -4, -4], [12, 5, 8, 0, 8, 6, 2, -8, -7, 7], [10, -6, 4, -10, -10, -4, -6, 14, -5, -11], [11, -2, -1, -1, 7, 5, -4, -1, -4, 2], [-11, 3, -8, -8, -4, -11, 3, 0, 1, 4], [13, 13, 15, 7, -3, 1, -16, 7, -16, -13], [9, 5, -5, -1, 10, 4, 1, -1, 1, -5], [-4, -1, 14, 11, 14, 2, 1, -14, 0, 0], [15, 7, 1, 5, 14, 0, -1, 5, -1, -7], [22, 5, -5, -10, -5, 13, -8, 5, 3, -3], [-12, 11, 0, 1, -5, -7, 7, 1, 7, -11], [-12, 6, 13, 19, 13, -6, 0, -13, -6, 6], [6, -9, 2, -9, 18, -14, 12, -9, 12, 9], [5, -2, 3, 6, 3, 8, -2, -3, 1, -1], [-15, 14, 13, 15, 13, 10, -12, -13, -2, 2], [0, 1, 1, 11, 12, 10, 1, -10, -2, -1], [13, -2, 11, 10, 3, 12, -2, 1, 0, -2], [19, 12, 1, 1, 1, 6, -6, -1, -6, 6], [-2, -4, -6, 5, 14, 9, -4, -11, 12, 8], [-7, -3, -10, -7, -11, -4, -3, -3, 13, 10], [7, -12, -4, -6, -4, -7, 9, 4, 3, -3], [9, -2, 7, 4, -10, 6, -2, 3, 5, 3], [12, 5, 17, 20, 30, 15, 5, -3, -5, 0], [-9, 12, 9, 8, 9, 3, -2, -9, -10, 10], [7, -4, 4, -3, 4, -4, 11, -4, -7, 7], [10, -1, 4, 7, 5, 1, -5, 7, -5, 1], [-8, 2, -6, 7, 28, 5, 2, -13, 5, 7], [-2, -14, 0, -4, 0, -9, 13, 0, 1, -1], [-27, 11, -7, 1, -7, -7, -1, 7, -10, 10], [18, 20, -3, 5, 4, 17, -14, 5, -14, -20], [-14, -11, -3, 6, -9, -2, 5, 6, 5, 11], [-16, -6, 7, -1, -9, -14, 7, -1, 7, 6], [-6, 6, 0, 6, -10, 0, 6, -6, -7, -1], [7, 0, -5, -10, -5, 7, -2, 5, 2, -2], [3, -4, -1, 1, -3, 5, -4, -2, -13, -17], [-9, -2, -7, -8, -7, 6, -5, 7, 7, -7], [-29, 3, -19, -9, -19, -10, 0, 19, -3, 3], [17, -2, 15, 7, 23, 9, -2, 8, -8, -10], [-7, -10, -14, -6, 8, -1, 15, -6, 15, 10], [-6, -1, 8, 7, -9, -5, -3, 7, -3, 1], [8, -6, 12, -1, 12, 9, 4, -12, 2, -2], [7, 11, 14, 4, -1, -6, -8, 4, -8, -11], [-19, 3, -7, -2, -7, -4, -1, 7, -2, 2], [24, -9, 15, 4, 16, 13, -9, 11, -6, -15], [-3, -3, 6, -14, 1, -10, 4, -14, 4, 3], [22, 16, -2, 0, -2, 9, -11, 2, -5, 5], [-4, 3, -1, -5, -24, -8, 3, 4, -8, -5], [-16, -15, 7, 5, 7, -10, 12, -7, 3, -3], [-20, 10, -10, 7, 8, -3, 10, -17, -4, 6], [-23, -7, -13, -5, -3, -7, 20, -5, 20, 7], [18, 6, -2, 1, -2, 7, -10, 2, 4, -4], [14, -9, -1, -7, 12, 3, -2, -7, -2, 9]];
    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_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_414_2_e_c(: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_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_414_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_414_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![25, -105, 326, -483, 554, -205, 136, -46, 25, -5, 1]>],Snew);
    return Vf;
end function;