
// Make newform 637.2.u.e in Magma, downloaded from the LMFDB on 04 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_u_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_637_2_u_e();".
// 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_637_2_u_e();".
// 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 := [4, -2, -1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, -1, 1, 1], [2, 1, 1, -1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_637_u();"
function MakeCharacter_637_u()
    N := 637;
    order := 6;
    char_gens := [248, 197];
    v := [2, 1];
    // 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_637_u_Hecke();"
function MakeCharacter_637_u_Hecke( : Kf := false)
    N := 637;
    order := 6;
    char_gens := [248, 197];
    char_values := [[-1, 0, 1, 0], [0, 0, 1, 0]];
    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], [0, 1, 0, 1], [1, -1, 0, 0], [0, 0, 0, 0], [-2, 1, 4, -1], [4, 0, -1, 0], [-3, 0, 3, 0], [0, 3, 0, -3], [-2, 4, 0, -2], [5, -1, -6, 2], [10, 0, -5, 0], [-8, 0, 4, 0], [2, 2, 4, 0], [3, -6, -7, 3], [3, 2, 5, 0], [4, -8, -3, 4], [3, -8, -5, 0], [2, -6, 0, -6], [-1, 6, 2, -6], [2, 0, -2, 2], [-4, 0, 2, 0], [6, 0, -6, 0], [3, -4, -6, 4], [11, 0, -8, 5], [-13, 0, 5, 3], [8, -1, 0, -1], [-1, 2, 3, -4], [1, -2, 6, 1], [0, 0, 3, -6], [2, -4, -9, 2], [2, 5, 3, -10], [12, -3, -15, 6], [7, -11, -4, 0], [1, -2, -3, 1], [0, -1, 0, 1], [-14, 0, 7, 0], [-5, 10, -3, -5], [-4, 0, 8, 0], [11, -2, 9, 0], [3, 6, 0, 6], [-9, 0, 0, 0], [2, -4, 0, -4], [-6, -3, 0, -3], [8, -12, -16, 12], [-5, 5, 0, 0], [11, 0, -11, 0], [-7, 2, 9, -4], [-8, 15, 7, 0], [-5, 8, 3, 0], [4, 0, 4, 0], [-7, 5, 12, -10], [-5, 10, 10, -10], [0, -9, -9, 0], [-4, 8, -6, -4], [5, -10, 9, 5], [-13, 8, 0, 8], [-14, 1, 15, -2], [-24, 0, 9, 6], [-1, 6, 7, -12], [-14, 14, 28, -14], [14, -6, 0, -6], [-6, 0, 2, 2], [5, -15, -10, 15], [-7, 14, 21, -7], [4, 6, 2, -12], [-7, 14, 7, 0], [-12, 9, 24, -9], [-4, 5, 0, 5], [10, -2, -12, 4], [-3, 12, 9, 0], [-6, 17, 12, -17], [-6, 5, -1, 0], [18, 0, 0, 0], [35, -1, 0, -1], [-5, -9, -14, 0], [1, 1, -2, -1], [-8, 16, -6, -8], [14, -18, -28, 18], [-1, 0, 12, -23], [1, 3, 4, 0], [9, 6, -3, -12], [-4, -6, 8, 6], [15, -19, -30, 19], [12, -24, -23, 12], [7, -14, -15, 7], [8, -4, -12, 8], [-26, 0, 14, -2], [-8, 0, 10, -12], [-13, 0, 4, 5], [-3, 6, 6, -6], [23, -4, -27, 8], [8, -11, -16, 11], [2, -15, -13, 0], [-12, -9, 3, 18], [-22, 9, -13, 0], [-9, 18, 12, -9], [4, 0, 5, -14], [2, -4, 18, 2], [-8, 16, 18, -8], [-15, 9, -6, 0], [5, -6, 0, -6], [-13, 23, 26, -23], [22, -8, -30, 16], [6, -12, -12, 6], [5, -10, 6, 5], [-32, 0, 19, -6], [-16, 17, 1, 0], [10, -5, 5, 0], [3, -6, 9, 3], [-7, 3, 10, -6], [3, 6, 0, 6], [-5, 21, 10, -21], [-26, 0, 11, 4], [-4, 0, 8, -12], [9, 0, -12, 15], [-24, 3, 27, -6], [-34, 0, 23, -12], [4, -14, 0, -14], [1, -2, -15, 1], [0, 0, -6, 0], [26, -12, -52, 12], [-5, 12, 17, -24], [-31, -1, 30, 2], [-15, -2, -17, 0], [11, 0, 2, -15], [14, -10, 0, -10], [8, -3, -16, 3], [-13, -4, 0, -4], [17, -6, 0, -6], [6, 3, 9, 0], [8, -12, -16, 12], [6, 11, 17, 0], [6, -12, -19, 6], [-6, 0, 6, 0], [3, -17, -6, 17], [13, -6, 7, 0], [-9, 16, 7, 0], [0, 12, 12, 0], [0, 0, -12, 0], [3, 18, 0, 18], [12, -15, -24, 15], [-36, 0, 11, 14], [1, -12, -13, 24], [17, -13, -34, 13], [-26, -3, 0, -3], [43, 0, -18, -7], [-5, 12, 10, -12], [7, -14, 21, 7], [-14, 0, 14, 0], [-6, -16, -22, 0], [-5, -12, 10, 12], [13, -2, -15, 4], [-21, 18, 0, 18], [2, -4, 24, 2], [-2, -14, 0, -14], [-18, -9, 0, -9], [-2, 10, 0, 10], [-25, 32, 50, -32], [13, -12, 0, -12], [24, 0, -19, 14], [55, 0, -24, -7], [0, 0, 15, 0], [9, -18, -18, 18], [-39, 6, 0, 6], [5, 22, -10, -22], [3, 0, 7, -17], [-32, -3, 0, -3], [11, 14, 3, -28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_637_u_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_u_e();".
// 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_637_2_u_e(:prec:=4)
    chi := MakeCharacter_637_u();
    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_637_2_u_e();".
// 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_637_2_u_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 2, 3, 1]>,<3,R![-5, -1, 1]>],Snew);
    return Vf;
end function;