
// Make newform 414.2.i.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_414_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_414_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_414_2_i_f();".
// 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_i_f();".
// 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, 1, -1, 1, -1, 1, -1, 1, -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_414_i();"
function MakeCharacter_414_i()
    N := 414;
    order := 11;
    char_gens := [47, 235];
    v := [11, 5];
    // 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_i_Hecke();"
function MakeCharacter_414_i_Hecke(Kf)
    N := 414;
    order := 11;
    char_gens := [47, 235];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 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 := [[0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -2, 1, 0, 0, -1, 0, 0, 1], [0, 0, -1, 0, -1, -1, -1, 0, -1, 0], [1, 2, -2, -1, 0, 0, 1, 1, 1, 0], [-2, 2, 0, 0, 0, 1, -2, 1, 0, 0], [-2, -1, -1, 3, -1, -1, -2, 0, 0, 0], [0, -1, -2, 2, 1, 0, 0, 1, 2, 1], [0, 0, 1, -2, -2, -1, 0, -1, -1, 3], [0, -1, -3, 0, -3, -1, 0, 0, -2, 2], [-5, 2, -5, 0, -3, 5, -3, 3, -5, 3], [5, -5, 1, 0, 0, -1, 5, -5, 0, -5], [7, -5, 4, -5, 7, 0, 5, -4, 4, -5], [-1, 3, 0, 3, -4, 4, -3, 0, -3, 1], [-1, 0, 0, -2, 4, 2, -2, -4, 2, 0], [0, 0, 0, 6, -3, 3, -3, 6, 0, 0], [-2, 2, 0, 2, -4, 4, -3, 4, -4, 2], [6, -5, 6, 0, 5, 1, 2, -2, -1, -5], [-2, 5, 0, -2, 1, -2, 0, 5, -2, 0], [-8, 3, -4, 2, -5, 2, -4, 3, -8, 0], [-2, 1, -3, 3, -1, 2, 0, 0, -2, 0], [-1, 1, 0, 2, -2, 2, -5, 2, -2, 2], [6, -7, 9, -5, 5, -9, 7, -6, 0, -5], [3, 1, 4, -4, 0, 0, 4, -4, -1, -3], [-3, 0, 3, 0, -3, 0, -2, -3, 3, 2], [2, -3, -3, 0, 0, 3, 3, -2, 0, -4], [1, -3, 3, -1, 0, -6, 2, -3, 2, -6], [0, -2, 0, 0, -2, 3, -6, 6, -3, 2], [-4, 3, -7, 6, -7, 3, -4, 0, -5, 5], [-2, 3, 3, 1, 2, 1, 3, 3, -2, 0], [3, -1, 1, -3, 0, -2, -1, 3, -1, -2], [0, -3, 2, -2, -2, -1, -2, -2, 2, -3], [2, 0, 3, -3, -2, -3, 3, 2, 3, -3], [-2, 0, 0, -10, 1, -5, 5, -1, 10, 0], [-8, 6, -6, 8, 0, 2, -8, 3, -8, 2], [-2, 2, 0, 6, -6, 5, -8, 5, -6, 6], [-4, 4, 1, -1, -4, 4, 0, -5, 8, -5], [-4, -9, 8, -8, 1, -8, 8, -9, -4, 0], [7, 0, 0, -6, 0, 0, 7, 0, 5, -5], [7, -2, 2, -7, 0, -10, 8, -14, 8, -10], [0, 2, 2, 2, 0, 0, -5, -4, 4, 5], [-6, 6, -2, 10, -6, 6, -10, 2, -6, 6], [0, 3, -6, 9, -4, -6, -4, 9, -6, 3], [-2, -12, 4, -9, 9, -4, 12, 2, 0, -5], [1, -1, 0, 0, -4, -4, -2, -4, -4, 0], [4, 3, 4, 0, -1, -7, -2, 2, 7, 1], [-13, 10, -11, 11, -10, 13, 0, 16, -18, 16], [0, 5, -8, -2, 5, 6, 5, -2, -8, 5], [-6, 1, 0, 6, -2, 2, -6, 0, -1, 6], [13, 0, 1, -5, 11, -9, 9, -11, 5, -1], [-8, -2, -1, -5, -5, 5, 5, 1, 2, 8], [2, 4, -3, -4, 4, 3, -4, -2, 0, 7], [1, -1, 1, -1, 0, -10, 2, -5, 2, -10], [-7, 4, -14, 4, -7, 4, -14, 4, -7, 0], [4, 3, -5, 5, -3, -4, 0, 1, -6, 1], [-8, 8, 0, 10, 1, 2, 5, 2, 1, 10], [0, -3, 7, 3, -2, 5, -2, 3, 7, -3], [-6, -5, 3, -5, -6, 0, 0, 2, -2, 0], [2, 0, -11, 10, -5, 12, -12, 5, -10, 11], [10, -5, 3, -5, 10, 0, -4, -6, 6, 4], [0, 0, 13, 4, -2, -6, -2, 4, 13, 0], [1, 6, 7, 1, 7, 6, 1, 0, 0, 0], [8, 4, 8, 0, 18, -14, 6, -6, 14, -18], [-2, 2, -2, 2, 0, 1, -1, 14, -1, 1], [-1, 0, 3, -3, 3, -3, 0, 1, 0, -7], [-4, -8, 14, -8, -4, 0, 2, 7, -7, -2], [1, 9, -8, -6, 6, 8, -9, -1, 0, 0], [-4, -14, -4, 0, 8, 1, -6, 6, -1, -8], [6, -20, 9, -12, 9, -12, 9, -20, 6, 0], [0, -9, -8, 8, 9, 0, 0, -2, 3, -2], [-3, 1, -3, 0, -6, 2, -6, 6, -2, 6], [-1, -6, -7, 9, -9, 7, 6, 1, 0, -3], [14, 0, 6, -9, -2, 1, -1, 2, 9, -6], [7, -5, 13, -5, 7, 0, -6, -4, 4, 6], [-10, 11, -11, 10, 0, 6, -18, 3, -18, 6], [14, -9, 14, 0, 8, -1, 6, -6, 1, -8], [18, -5, 10, -1, 4, -1, 10, -5, 18, 0], [0, -2, -3, 1, -3, -2, 0, 0, -14, 14], [4, -4, 0, -14, 10, 0, 4, 0, 10, -14], [9, -13, 19, -13, 9, 0, 4, -10, 10, -4], [-3, -8, -4, -8, -3, 0, -5, 10, -10, 5], [0, 5, -14, 12, -14, 5, -14, 12, -14, 5], [-13, -1, -7, -7, -7, -1, -13, 0, 3, -3], [-6, -3, 2, -2, 3, 6, 0, -4, 10, -4], [4, -11, 13, -5, 22, -5, 13, -11, 4, 0], [1, 6, 1, -4, 1, 6, 1, 0, 0, 0], [1, -4, 4, -1, 0, -15, -3, -11, -3, -15], [-12, 10, -8, -11, -5, 5, 11, 8, -10, 12], [-11, 0, -24, 19, -11, 23, -23, 11, -19, 24], [-5, 2, 4, -2, 2, -2, 2, -4, -2, 5], [0, 6, 3, -4, 7, 11, 7, -4, 3, 6], [11, -2, 4, -4, 2, -11, 0, -4, 17, -4], [12, -17, 7, -7, 17, -12, 0, -10, 11, -10], [3, 2, 3, 0, 7, -7, -9, 9, 7, -7], [0, 10, -8, 14, -19, 3, -19, 14, -8, 10], [-14, 6, 7, 3, -5, 5, -3, -7, -6, 14], [-8, 24, -5, 6, -3, 3, -6, 5, -24, 8], [-8, 6, -8, 0, -5, 6, -19, 19, -6, 5], [-14, 10, -12, 4, -12, 10, -14, 0, 3, -3], [7, -1, 1, -7, 0, -8, 8, 1, 8, -8], [-5, -11, 7, -11, -5, 0, 7, 1, -1, -7], [2, -6, -17, -6, 2, 0, -9, 0, 0, 9], [16, -9, 9, -16, 0, -15, 16, -17, 16, -15], [10, -14, 2, -9, 2, -14, 10, 0, 9, -9], [-6, -5, -28, 28, 5, 6, 0, 10, 1, 10], [3, 5, -5, -14, 3, -14, -5, 5, 3, 0], [-12, 8, -8, 12, 0, 9, -10, 4, -10, 9], [-4, 16, 0, 16, -4, 0, 8, -1, 1, -8], [-5, 0, 6, -2, -10, 4, -4, 10, 2, -6], [-11, 3, -15, -3, -3, 3, 3, 15, -3, 11], [22, -4, 6, 7, -7, -6, 4, -22, 0, -8], [17, -9, 17, 0, 4, -9, 3, -3, 9, -4], [4, -10, -4, 4, 10, -4, 0, -16, 1, -16], [-5, 13, -1, 2, 2, 2, -1, 13, -5, 0], [-1, 1, 0, -2, 5, -14, -2, -14, 5, -2], [-4, 12, -11, 7, -23, 23, -7, 11, -12, 4], [29, 0, 5, 0, 12, -10, 10, -12, 0, -5], [-17, -5, -4, 11, 0, 0, -11, 4, 5, 17], [2, 14, 0, 8, -8, 0, -14, -2, 0, -2], [-9, 14, -9, 0, -8, 4, 6, -6, -4, 8], [1, 15, -17, 7, -17, 15, 1, 0, -14, 14], [-4, -4, 0, -23, 0, -4, -4, 0, 1, -1], [-6, 6, 0, -5, 15, -17, -3, -17, 15, -5], [0, 11, 9, 11, 0, -2, 0, 11, 9, 11], [3, 0, 7, -8, -8, -13, 13, 8, 8, -7], [3, -1, 1, -3, 0, 19, -3, 6, -3, 19], [13, -21, 27, -27, 21, -13, 0, -14, 10, -14], [21, -16, 15, -9, 15, -16, 21, 0, 7, -7], [-18, 7, -6, 3, -3, 6, -7, 18, 0, -15], [-18, 12, -1, 14, -6, 6, -14, 1, -12, 18], [-6, -5, -2, 8, -4, 4, -8, 2, 5, 6], [0, 0, -3, 5, 8, -26, 8, 5, -3, 0], [-4, 10, -4, 0, -11, 18, -4, 4, -18, 11], [5, -8, 9, -10, 34, -10, 9, -8, 5, 0], [-18, 21, -6, -1, -3, -1, -6, 21, -18, 0], [-3, 3, 0, -5, -2, -14, 16, -14, -2, -5], [14, -6, -2, -16, 16, 2, 6, -14, 0, -5], [20, -12, 27, -12, 20, 0, -2, -15, 15, 2], [19, -8, 19, 0, 16, -7, 17, -17, 7, -16], [1, -11, 11, -11, 11, -1, 0, -11, -9, -11], [16, 11, -4, -2, -4, 11, 16, 0, -3, 3], [0, 19, -21, 12, -30, 20, -30, 12, -21, 19], [9, 5, -16, 5, 9, 0, 7, -2, 2, -7], [-6, 0, 20, 0, 13, -7, 7, -13, 0, -20], [-11, 0, -14, 11, -8, 25, -25, 8, -11, 14], [-7, 1, -1, 7, 0, -8, 0, -29, 0, -8], [-13, 9, 7, -2, -11, -2, 7, 9, -13, 0], [9, -15, 14, -17, 14, -15, 9, 0, 14, -14], [15, -3, 15, 0, 11, 0, 1, -1, 0, -11], [10, 1, -1, -10, 0, 8, -5, 9, -5, 8], [11, -9, 20, 4, -4, 4, -4, -20, 9, -11], [0, -5, 14, 14, 1, -8, 1, 14, 14, -5], [9, -8, 18, -9, 9, -18, 8, -9, 0, -15], [-5, 5, 0, -2, -9, 11, -6, 11, -9, -2], [-10, 10, 0, -6, 7, 6, 12, 6, 7, -6], [16, -12, 14, -29, 29, -14, 12, -16, 0, -35], [-26, 0, -15, 27, -22, 13, -13, 22, -27, 15], [-23, 12, -28, 5, -5, 28, -12, 23, 0, 13], [-17, -1, -6, -8, -6, -1, -17, 0, -8, 8], [-19, 8, 2, 6, 22, 6, 2, 8, -19, 0], [3, 20, -6, 6, -20, -3, 0, 3, -16, 3], [-7, 7, 0, 18, 0, -1, -4, -1, 0, 18], [-8, 0, 20, -8, 15, -5, 5, -15, 8, -20], [2, 32, 0, 32, 2, 0, -9, 7, -7, 9], [5, -4, 4, -5, 0, 0, -11, -5, -11, 0], [3, 17, -9, -18, -9, 17, 3, 0, 17, -17], [4, 7, -11, -14, -9, -14, -11, 7, 4, 0], [20, -16, 20, 0, 3, -26, 2, -2, 26, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_414_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_414_2_i_f();".
// 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_i_f(:prec:=10)
    chi := MakeCharacter_414_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_414_2_i_f();".
// 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_i_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_414_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![529, 713, 587, 201, 126, -153, 86, -29, 14, -6, 1]>],Snew);
    return Vf;
end function;