
// Make newform 213.2.e.a in Magma, downloaded from the LMFDB on 03 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_213_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_213_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_213_2_e_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_213_2_e_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];
        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_213_e();"
function MakeCharacter_213_e()
    N := 213;
    order := 5;
    char_gens := [143, 7];
    v := [5, 4];
    // 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_213_e_Hecke();"
function MakeCharacter_213_e_Hecke( : Kf := false)
    N := 213;
    order := 5;
    char_gens := [143, 7];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1], [1, -1, 1, -1], [0, -3, 2, -3], [1, 0, 0, -1], [-3, -2, 2, 3], [2, -1, 2, 0], [0, 0, 0, 0], [4, 1, -1, -4], [-1, 0, -4, 4], [0, 7, -2, 7], [-5, 1, -1, 5], [-3, 0, 1, -1], [0, 0, 5, -5], [0, 1, -2, 1], [-6, 3, -6, 0], [0, -1, -11, -1], [0, 3, 1, 3], [1, -8, 1, 0], [4, -4, 0, 4], [-3, -1, -1, -7], [6, 1, -1, -6], [0, -8, 11, -8], [0, -8, 1, -8], [0, 3, -8, 3], [-1, 0, -8, 8], [12, 0, 9, -9], [-10, 0, 3, -3], [15, -4, 15, 0], [-1, 1, 0, -3], [0, 4, -15, 4], [-4, -1, -4, 0], [-10, -4, 4, 10], [0, 12, -13, 12], [-8, 16, -16, 8], [4, -8, 8, -4], [-10, 10, 0, 0], [15, -7, 15, 0], [2, -2, 0, 6], [4, -11, 11, -4], [1, 7, -7, -1], [1, 0, 10, -10], [7, 0, 11, -11], [-12, 12, 0, 4], [2, 0, 11, -11], [-6, 7, -6, 0], [-15, 15, 0, 4], [-18, 16, -16, 18], [-6, 4, -6, 0], [-5, 5, 0, 0], [-9, 7, -9, 0], [2, 0, 5, -5], [16, 0, 5, -5], [0, -2, 21, -2], [3, -3, 0, 9], [0, 6, -4, 6], [7, -7, 0, -14], [8, -1, 8, 0], [-4, -11, -4, 0], [-14, 21, -21, 14], [15, -12, 12, -15], [13, 0, 10, -10], [7, -7, 0, -11], [12, 0, 18, -18], [0, -5, -27, -5], [0, 15, -11, 15], [-3, 3, 0, -20], [6, 25, 6, 0], [0, 1, -22, 1], [0, 6, 4, 6], [22, -22, 0, -3], [-25, 17, -17, 25], [-6, 6, 0, 29], [0, 4, 1, 4], [0, -12, 22, -12], [12, -22, 12, 0], [0, 0, 5, 0], [-2, 0, 11, -11], [0, 16, -2, 16], [-6, -10, 10, 6], [-8, 20, -8, 0], [1, 13, -13, -1], [0, -5, -30, -5], [0, -18, 7, -18], [-17, 2, -2, 17], [-6, -22, -6, 0], [-19, 7, -19, 0], [15, 0, 7, -7], [-5, -3, 3, 5], [-2, 19, -2, 0], [0, 0, -10, 10], [33, 0, 7, -7], [0, 2, -3, 2], [-22, 7, -22, 0], [-24, 24, 0, 20], [5, -10, 10, -5], [23, -23, 0, -9], [0, -20, 13, -20], [2, -15, 2, 0], [15, 0, 33, -33], [0, -3, 10, -3], [-13, 13, 0, 9], [24, -35, 35, -24], [0, 7, -5, 7], [8, 0, 13, -13], [-14, -12, 12, 14], [-1, 1, 0, -24], [20, -24, 24, -20], [11, 19, -19, -11], [1, -19, 19, -1], [16, -16, 0, 6], [18, 0, 5, -5], [-17, 0, 6, -6], [-8, 8, 0, -4], [-16, 0, -20, 20], [0, -14, -25, -14], [-13, 1, -1, 13], [-6, 6, 0, 19], [0, -10, -6, -10], [-9, 9, 0, 35], [-16, 0, -1, 1], [8, -8, 0, -18], [-4, 0, -20, 20], [-30, 30, 0, 8], [0, -27, 17, -27], [32, -7, 7, -32], [9, -9, 0, 31], [-13, 0, -6, 6], [4, -4, 0, 8], [6, 3, 6, 0], [-23, 0, -16, 16], [0, 12, 28, 12], [-26, 26, 0, 17], [26, 0, 22, -22], [10, 30, 10, 0], [16, 0, 5, -5], [0, -17, -8, -17], [0, 12, -8, 12], [14, -14, 0, -34], [34, -8, 34, 0], [0, 3, -9, 3], [0, 0, -24, 24], [-7, -8, 8, 7], [0, -11, 18, -11], [34, -31, 31, -34], [-11, 0, 18, -18], [-30, 1, -30, 0], [-1, 0, -7, 7], [0, 13, 21, 13], [12, -22, 22, -12], [35, -22, 22, -35], [-15, 20, -20, 15], [0, -1, 51, -1], [-5, -12, 12, 5], [6, -10, 6, 0], [-16, -8, -16, 0], [0, -2, -27, -2], [-20, 20, 0, 19], [-9, 9, 0, 50], [42, -42, 0, -24], [0, -5, -2, -5], [-14, 36, -14, 0], [22, 0, 11, -11], [0, 13, -6, 13], [-5, 0, 8, -8], [19, 12, 19, 0], [17, -3, 3, -17], [-21, 19, -19, 21], [-6, 5, -5, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_213_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_213_2_e_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_213_2_e_a(:prec:=4)
    chi := MakeCharacter_213_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_213_2_e_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_213_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_213_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 4, -3, 1]>],Snew);
    return Vf;
end function;