
// Make newform 414.2.i.a in Magma, downloaded from the LMFDB on 16 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_a();".
// 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_a();".
// 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], [-1, 0, -1, 0, -1, 0, -1, 2, -2, 1], [0, -1, 0, 1, -1, 0, -1, 1, 0, -1], [1, 0, 0, -1, 0, -2, 0, 0, 0, -2], [3, -3, 0, -2, 2, -3, 0, -3, 2, -2], [1, 0, 1, 1, 1, 0, 1, 0, 4, -4], [-1, 0, -1, 1, 0, 1, 0, 2, -3, 2], [4, -1, 0, -3, 1, -2, 1, -1, -1, 0], [-2, -1, 0, -5, 0, -1, -2, 0, 1, -1], [-4, 4, -4, 0, -3, 5, -4, 4, -5, 3], [-1, -2, 0, -1, 1, 0, 2, 1, 0, 4], [3, -1, 1, -1, 3, 0, 2, -5, 5, -2], [2, 1, -4, 1, -2, 2, -1, 4, -1, -2], [1, 0, 1, -1, 2, -5, 5, -2, 1, -1], [0, -4, 0, 2, 2, -1, 2, 2, 0, -4], [-1, 1, 0, 6, -5, 2, -4, 2, -5, 6], [-1, 7, -1, 0, -2, -2, 1, -1, 2, 2], [-1, -6, 4, -1, 1, -1, 4, -6, -1, 0], [-1, -5, 3, 3, 3, 3, 3, -5, -1, 0], [9, -4, 5, -5, 4, -9, 0, -6, 8, -6], [1, -1, 0, 1, 6, -7, 1, -7, 6, 1], [-4, 5, -6, 4, -4, 6, -5, 4, 0, 9], [1, -1, 3, 4, -1, 1, -4, -3, 1, -1], [3, 5, -4, 5, 3, 0, 1, 0, 0, -1], [8, 0, 3, 2, -2, -3, 0, -8, 0, 1], [-1, -1, 1, 1, 0, 8, 0, 3, 0, 8], [-6, 5, -6, 0, -2, 0, -2, 2, 0, 2], [0, 6, -6, 13, -6, 6, 0, 0, -6, 6], [1, -1, -5, 2, -4, 2, -5, -1, 1, 0], [-2, -3, 3, 2, 0, -2, -7, 2, -7, -2], [0, -3, 11, 0, 6, -1, 6, 0, 11, -3], [0, 0, 5, -9, 0, -3, 3, 0, 9, -5], [-2, 0, -3, 10, 0, 2, -2, 0, -10, 3], [4, -1, 1, -4, 0, 1, 2, -6, 2, 1], [-6, 6, 0, 2, 3, 3, 4, 3, 3, 2], [2, -7, 11, -11, 7, -2, 0, -11, -2, -11], [-2, 5, -7, 1, -4, 1, -7, 5, -2, 0], [-1, 7, -1, -3, -1, 7, -1, 0, -2, 2], [5, 5, -5, -5, 0, 4, -8, 2, -8, 4], [-1, -3, -4, -3, -1, 0, 0, 3, -3, 0], [20, -16, 18, -21, 17, -17, 21, -18, 16, -20], [0, -12, 1, -2, 8, -9, 8, -2, 1, -12], [-2, -1, -1, 5, -5, 1, 1, 2, 0, -6], [-10, 10, 0, 2, -4, -3, -1, -3, -4, 2], [4, 9, 4, 0, -5, 7, -3, 3, -7, 5], [1, 6, 8, -8, -6, -1, 0, -7, 0, -7], [0, 0, 5, 6, 5, -4, 5, 6, 5, 0], [2, 1, 4, 3, 3, -3, -3, -4, -1, -2], [-11, 0, 4, -5, 2, 1, -1, -2, 5, -4], [8, 1, -5, -4, 0, 0, 4, 5, -1, -8], [-9, 3, -7, 13, -13, 7, -3, 9, 0, 8], [1, 3, -3, -1, 0, -3, 4, 16, 4, -3], [-3, 4, 0, 13, 2, 13, 0, 4, -3, 0], [16, -10, 7, -7, 10, -16, 0, 1, 8, 1], [13, -13, 0, -8, 3, -9, 7, -9, 3, -8], [0, 0, 8, -13, 5, 4, 5, -13, 8, 0], [-10, 6, -11, 6, -10, 0, -15, -1, 1, 15], [-4, 0, -2, 7, 1, 10, -10, -1, -7, 2], [3, 3, 12, 3, 3, 0, -4, -1, 1, 4], [0, -2, -11, -1, 0, 4, 0, -1, -11, -2], [3, 8, -11, 6, -11, 8, 3, 0, -11, 11], [-9, 2, -9, 0, 5, 3, -2, 2, -3, -5], [0, 1, -1, 0, 0, 2, -7, -4, -7, 2], [-1, -1, 10, 0, 0, -10, 1, 1, 0, 2], [5, 3, 6, 3, 5, 0, 1, -5, 5, -1], [4, -7, 10, 6, -6, -10, 7, -4, 0, 4], [-8, 4, -8, 0, -13, 2, -2, 2, -2, 13], [-8, -2, -5, -7, 5, -7, -5, -2, -8, 0], [-4, 1, -4, 4, -1, 4, 0, 0, 11, 0], [-4, 5, -4, 0, -2, -3, 1, -1, 3, 2], [-5, 1, 7, -1, 1, -7, -1, 5, 0, -5], [1, 0, 0, -2, -8, 7, -7, 8, 2, 0], [2, -5, -3, -5, 2, 0, -5, 11, -11, 5], [6, -3, 3, -6, 0, 6, 2, 10, 2, 6], [-3, 16, -3, 0, -19, 14, -13, 13, -14, 19], [-9, 6, -8, 13, 0, 13, -8, 6, -9, 0], [-8, 9, -10, 12, -10, 9, -8, 0, -8, 8], [-7, 7, 0, 8, -11, -10, 9, -10, -11, 8], [8, 13, -8, 13, 8, 0, -6, 1, -1, 6], [7, -1, 0, -1, 7, 0, -9, -3, 3, 9], [0, -4, 7, -18, 16, 9, 16, -18, 7, -4], [6, 3, 1, -12, 1, 3, 6, 0, -4, 4], [-6, 10, -2, 2, -10, 6, 0, 12, -8, 12], [3, -2, -14, 4, -1, 4, -14, -2, 3, 0], [9, -12, -2, -3, -2, -12, 9, 0, 0, 0], [-8, 4, -4, 8, 0, -1, 4, -12, 4, -1], [5, -4, 7, -6, 2, -2, 6, -7, 4, -5], [-18, 0, 0, 3, -2, 4, -4, 2, -3, 0], [-4, -1, -18, -2, -1, 1, 2, 18, 1, 4], [0, -6, 13, 1, 0, -1, 0, 1, 13, -6], [-5, 1, 5, -5, -1, 5, 0, 0, -11, 0], [0, 0, 7, -7, 0, 0, 0, -14, -1, -14], [0, 16, 0, 0, 0, -13, 0, 0, 13, 0], [0, 3, 10, -2, 2, -2, 2, -2, 10, 3], [6, -21, 6, -15, 13, -13, 15, -6, 21, -6], [15, -8, 12, -15, -1, 1, 15, -12, 8, -15], [-12, 7, -12, 0, -2, 5, -7, 7, -5, 2], [7, -2, 16, -22, 16, -2, 7, 0, 6, -6], [-16, -4, 4, 16, 0, 5, -14, 11, -14, 5], [-8, -7, -4, -7, -8, 0, -12, -7, 7, 12], [8, -8, 17, -8, 8, 0, 10, -1, 1, -10], [5, -4, 4, -5, 0, 8, 4, 13, 4, 8], [16, -9, 13, -5, 13, -9, 16, 0, 1, -1], [10, -2, -6, 6, 2, -10, 0, 3, 9, 3], [-4, 2, -7, -14, 11, -14, -7, 2, -4, 0], [-11, 11, -11, 11, 0, -1, 0, 7, 0, -1], [-3, 7, -1, 7, -3, 0, -19, 17, -17, 19], [-4, 0, 11, 9, 0, 6, -6, 0, -9, -11], [5, 1, 3, 4, 13, -13, -4, -3, -1, -5], [-5, 8, 17, 0, 0, -17, -8, 5, 0, 1], [-9, 8, -9, 0, -9, 4, 2, -2, -4, 9], [10, -20, 14, -14, 20, -10, 0, -18, -4, -18], [-19, 15, -22, 22, -20, 22, -22, 15, -19, 0], [17, -17, 0, 4, 9, 2, 13, 2, 9, 4], [-5, 3, -12, -6, -5, 5, 6, 12, -3, 5], [14, 0, 6, -17, 3, 0, 0, -3, 17, -6], [13, -5, 17, -8, 21, -21, 8, -17, 5, -13], [-6, 0, -5, -8, 8, 5, 0, 6, 0, -6], [5, 1, 5, 0, 5, 4, 1, -1, -4, -5], [9, 2, -5, 5, -5, 2, 9, 0, 4, -4], [-3, 0, 15, -13, 15, 0, -3, 0, 8, -8], [-2, 2, 0, 5, 9, -4, -18, -4, 9, 5], [0, 14, -21, 2, 4, 9, 4, 2, -21, 14], [9, 0, 24, -9, -1, 0, 0, 1, 9, -24], [-7, -17, 17, 7, 0, -5, 11, -12, 11, -5], [-1, -4, 10, -10, 4, 1, 0, -16, 1, -16], [-3, -14, 16, -11, 16, -14, -3, 0, 8, -8], [-3, 5, -14, 9, -9, 14, -5, 3, 0, 0], [-22, -4, -13, 23, -5, 5, -23, 13, 4, 22], [18, -11, 27, -20, 13, -13, 20, -27, 11, -18], [0, 14, 11, 0, -3, 14, -3, 0, 11, 14], [-1, -21, -1, 0, 0, -8, 13, -13, 8, 0], [2, -6, 10, 2, 2, 2, 10, -6, 2, 0], [7, 13, -6, 9, 1, 9, -6, 13, 7, 0], [-7, 7, 0, -13, 5, -1, 16, -1, 5, -13], [19, -23, 10, -26, 26, -10, 23, -19, 0, -6], [-23, 9, 9, 9, -23, 0, -17, 19, -19, 17], [3, 2, 3, 0, 1, 11, 17, -17, -11, -1], [11, 4, -4, 4, -4, -11, 0, 2, 0, 2], [1, 5, 10, 0, 10, 5, 1, 0, -14, 14], [0, 5, 3, 11, -26, 4, -26, 11, 3, 5], [5, 16, -23, 16, 5, 0, 8, 3, -3, -8], [11, 0, -12, 16, -9, 16, -16, 9, -16, 12], [-1, 0, 3, 8, 1, 13, -13, -1, -8, -3], [29, -2, 2, -29, 0, -4, 1, -17, 1, -4], [-12, -2, 12, 23, -7, 23, 12, -2, -12, 0], [5, -15, 12, -10, 12, -15, 5, 0, 17, -17], [-9, 4, -9, 0, -12, 13, -21, 21, -13, 12], [-18, 2, -2, 18, 0, 11, -12, 12, -12, 11], [-24, 25, -29, 16, -22, 22, -16, 29, -25, 24], [0, 5, 17, -12, -3, -12, -3, -12, 17, 5], [27, 1, 11, 7, -7, -11, -1, -27, 0, 2], [-5, 5, 0, 11, 10, 16, -17, 16, 10, 11], [19, -19, 0, -15, 11, -7, 0, -7, 11, -15], [-3, 15, -10, -1, 1, 10, -15, 3, 0, -8], [21, 0, 16, -10, 19, -21, 21, -19, 10, -16], [-12, 32, -16, 21, -21, 16, -32, 12, 0, 15], [-24, 0, -18, 25, -18, 0, -24, 0, 6, -6], [-22, 7, 3, -10, 0, -10, 3, 7, -22, 0], [-21, 11, -14, 14, -11, 21, 0, 1, -24, 1], [-13, 13, 0, 32, -6, 7, -3, 7, -6, 32], [29, 0, -4, -11, 22, -24, 24, -22, 11, 4], [5, 4, -5, 4, 5, 0, 2, 1, -1, -2], [1, -13, 13, -1, 0, -12, -4, -21, -4, -12], [-19, 6, -15, 7, -15, 6, -19, 0, -10, 10], [25, -15, 11, -5, -7, -5, 11, -15, 25, 0], [11, 28, 11, 0, 6, -3, 10, -10, 3, -6]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : 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, 138, 2324, 2185, 460, 76, 137, 41, 4, 2, 1]>],Snew);
    return Vf;
end function;