
// Make newform 104.2.s.a in Magma, downloaded from the LMFDB on 19 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_104_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_104_s_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_104_2_s_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_104_2_s_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, 0, -1, 0, 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_104_s();"
function MakeCharacter_104_s()
    N := 104;
    order := 6;
    char_gens := [79, 53, 41];
    v := [6, 3, 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_104_s_Hecke();"
function MakeCharacter_104_s_Hecke( : Kf := false)
    N := 104;
    order := 6;
    char_gens := [79, 53, 41];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 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], [-1, 1, -1, 0], [-2, -2, 0, 1], [-4, 0, 2, 0], [-2, 0, 2, 0], [0, 3, 0, -4], [0, 2, -3, 2], [0, 1, -3, 1], [-3, -3, 3, 6], [0, -3, 0, 0], [1, 0, -2, -3], [-6, 1, 6, -2], [-3, -4, -3, 0], [4, -6, -2, 6], [1, 0, -2, -5], [4, 0, -8, 3], [0, 2, -4, 2], [0, -1, 0, 1], [-9, 1, 9, -2], [-2, 11, 1, -11], [-1, 0, 2, 0], [0, -12, 0, 6], [2, -4, 0, 2], [2, 4, 2, 0], [0, -6, 0, 6], [-2, 15, -2, 0], [1, 6, 0, -3], [-5, -9, -5, 0], [-6, 0, 0, 0], [0, 0, 3, 0], [-9, 5, 9, -10], [1, 0, -2, -9], [-18, 2, 9, -2], [-4, -6, 2, 6], [0, 1, 10, 1], [3, 0, -6, 3], [-4, 0, 8, -3], [0, 6, 6, 6], [-4, 2, -4, 0], [8, 12, -4, -12], [-8, 0, -8, 0], [2, 0, -4, -15], [0, 3, -9, 3], [11, -6, 11, 0], [16, -2, -16, 4], [0, 7, 3, 7], [9, 13, 9, 0], [-13, -3, -13, 0], [0, -11, -7, -11], [-6, -8, 0, 4], [0, 0, 0, 0], [-6, 0, 12, 8], [6, 6, -3, -6], [6, 3, -3, -3], [-9, -8, 9, 16], [-6, -6, 6, 12], [-4, -6, 2, 6], [-10, 0, -10, 0], [-20, 3, 10, -3], [5, 0, -10, -2], [-12, -10, -12, 0], [0, 3, -8, 3], [3, -6, 0, 3], [-3, -18, 0, 9], [-6, 4, 0, -2], [-20, -6, 0, 3], [0, 0, -12, 0], [-9, -28, 0, 14], [-2, 3, 1, -3], [12, -10, -12, 20], [1, -20, 1, 0], [4, 0, -8, 20], [23, 3, -23, -6], [-24, -5, 12, 5], [3, 11, -3, -22], [0, 20, 0, -20], [-16, 0, 32, -9], [0, 14, 0, 14], [1, 8, 1, 0], [-6, 6, 3, -6], [13, -3, 13, 0], [18, 14, 0, -7], [8, 4, 8, 0], [0, 0, -21, 0], [-26, 6, 26, -12], [0, 0, 0, 0], [20, 4, -10, -4], [1, 6, 1, 0], [0, 5, 20, 5], [-6, 0, 12, 6], [2, 0, -4, 24], [1, -31, 1, 0], [20, 6, -10, -6], [-6, 18, -6, 0], [-21, 14, 0, -7], [0, -12, 12, -12], [-20, 7, 20, -14], [-15, 28, 0, -14], [4, -6, 4, 0], [-6, -34, 0, 17], [9, 0, -18, -3], [10, -3, -10, 6], [16, -30, -8, 30], [24, 0, -24, 0], [-6, 0, 12, -8], [1, 0, -2, 36], [14, 18, -14, -36], [9, 0, -18, 14], [-15, 2, 0, -1], [-33, 4, 33, -8], [0, -9, 13, -9], [-18, -13, 18, 26], [2, 16, -1, -16], [0, 28, 0, -14], [8, -24, -4, 24], [0, -4, 21, -4], [0, 12, 18, 12], [-21, -5, 21, 10], [0, 36, 0, 0], [-8, -24, 4, 24], [0, -7, 0, 14], [11, 0, -11, 0], [-8, 0, 16, -6], [0, 1, -13, 1], [-18, 0, 18, 0], [12, 0, -24, 12], [0, -3, -36, -3], [0, 16, 6, 16], [-24, 12, 0, -6], [-24, 22, 0, -11], [-12, -4, 12, 8], [2, 2, 2, 0], [26, 6, -26, -12], [18, -10, 18, 0], [4, -16, -2, 16], [6, 12, 6, 0], [0, -28, -4, -28], [0, 1, 39, 1], [-28, 6, 14, -6], [21, -16, -21, 32], [-3, -34, 0, 17], [34, 0, -34, 0], [0, -24, 10, -24], [-14, 48, 0, -24], [0, -19, 0, 0], [-34, -5, 17, 5], [18, 6, 0, -3], [-27, 20, 0, -10], [-3, 0, 6, -25], [-10, 0, 20, -8], [0, -3, 0, -3], [-39, 6, 39, -12], [3, 0, -6, -37], [24, 6, -24, -12], [18, -26, 18, 0], [-6, 56, 0, -28], [0, 13, 3, 13], [-6, 32, 3, -32], [-9, -40, 0, 20], [-28, 24, 0, -12], [5, -31, -5, 62], [0, -4, -18, -4], [7, 0, -14, 21], [-18, 9, 9, -9], [13, 10, 13, 0], [-27, 0, 54, 13], [-7, -9, 7, 18], [44, 15, -22, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_104_s_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_104_2_s_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_104_2_s_a(:prec:=4)
    chi := MakeCharacter_104_s();
    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_104_2_s_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_104_2_s_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_104_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 12, 14, 6, 1]>],Snew);
    return Vf;
end function;