
// Make newform 1568.2.i.o in Magma, downloaded from the LMFDB on 03 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1568_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1568_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_1568_2_i_o();".
// 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_1568_2_i_o();".
// 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, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 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_1568_i();"
function MakeCharacter_1568_i()
    N := 1568;
    order := 3;
    char_gens := [1471, 197, 1473];
    v := [3, 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_1568_i_Hecke();"
function MakeCharacter_1568_i_Hecke( : Kf := false)
    N := 1568;
    order := 3;
    char_gens := [1471, 197, 1473];
    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 := [[0, 0, 0, 0], [-1, 1, -1, 0], [0, 2, -1, 2], [0, 0, 0, 0], [-1, -1, -1, 0], [0, 0, 0, 2], [-3, 2, -3, 0], [0, 1, 5, 1], [0, 3, 1, 3], [0, 0, 0, 2], [-7, 1, -7, 0], [0, 4, 3, 4], [4, 0, 0, -2], [-4, 0, 0, 4], [0, 1, 9, 1], [1, 0, 1, 0], [-1, -7, -1, 0], [0, -4, -3, -4], [-7, 3, -7, 0], [-8, 0, 0, 4], [-9, -4, -9, 0], [0, -5, 1, -5], [4, 0, 0, 8], [0, 0, 9, 0], [4, 0, 0, 2], [5, 6, 5, 0], [0, 1, 7, 1], [0, 7, 3, 7], [11, -2, 11, 0], [4, 0, 0, -10], [0, 0, 0, -4], [0, 9, -5, 9], [3, -10, 3, 0], [4, 0, 0, -8], [0, -2, 9, -2], [-9, 7, -9, 0], [15, -4, 15, 0], [0, 3, -11, 3], [2, 0, 0, 0], [0, 0, -11, 0], [7, -1, 7, 0], [-2, 0, 0, -8], [0, 7, -1, 7], [-5, 10, -5, 0], [-16, 0, 0, 2], [-7, -3, -7, 0], [12, 0, 0, 0], [8, 0, 0, 4], [9, 5, 9, 0], [0, -2, 7, -2], [0, -2, 5, -2], [10, 0, 0, 8], [-1, 8, -1, 0], [-14, 0, 0, 12], [0, 6, -13, 6], [1, 7, 1, 0], [7, 4, 7, 0], [0, -15, -9, -15], [1, -8, 1, 0], [-4, 0, 0, 6], [-7, 1, -7, 0], [-6, 0, 0, 16], [20, 0, 0, 4], [23, -3, 23, 0], [0, 20, 1, 20], [0, 8, -21, 8], [0, -1, -27, -1], [12, 0, 0, 2], [25, 7, 25, 0], [-8, 0, 0, -2], [5, -2, 5, 0], [0, -13, 7, -13], [-17, 9, -17, 0], [0, -8, 3, -8], [18, 0, 0, -12], [0, -3, -7, -3], [-21, 2, -21, 0], [0, -2, 7, -2], [0, 2, 5, 2], [-11, 18, -11, 0], [4, 0, 0, -4], [-8, 0, 0, 6], [-17, 3, -17, 0], [-20, 0, 0, -6], [0, 21, -7, 21], [0, -5, 19, -5], [4, 0, 0, 2], [0, 20, -9, 20], [0, 0, 0, -18], [0, 0, 0, -8], [0, -11, 3, -11], [-1, 21, -1, 0], [1, -9, 1, 0], [2, 0, 0, 4], [0, 11, -11, 11], [-16, 0, 0, -4], [0, -6, -33, -6], [7, 0, 7, 0], [0, 9, 5, 9], [0, -2, -23, -2], [6, 0, 0, 12], [3, -2, 3, 0], [7, 5, 7, 0], [0, 4, 31, 4], [-17, -1, -17, 0], [39, 0, 39, 0], [12, 0, 0, -12], [0, -6, 3, -6], [9, 19, 9, 0], [-12, 0, 0, 10], [0, 13, 17, 13], [-13, 6, -13, 0], [-12, 0, 0, -22], [-25, 5, -25, 0], [-24, 0, 0, 4], [17, 0, 17, 0], [28, 0, 0, 12], [-1, -19, -1, 0], [0, -24, -5, -24], [-12, 0, 0, 16], [-25, -4, -25, 0], [12, 0, 0, 10], [0, 14, -1, 14], [-33, 11, -33, 0], [0, 17, 3, 17], [14, 0, 0, 0], [0, -28, 3, -28], [0, 17, 7, 17], [40, 0, 0, 4], [0, 8, -35, 8], [23, -21, 23, 0], [-32, 0, 0, -4], [0, -9, -9, -9], [8, 0, 0, 10], [0, 22, 11, 22], [-28, 0, 0, -10], [5, -10, 5, 0], [-25, -15, -25, 0], [-16, 0, 0, 22], [1, 12, 1, 0], [-12, 0, 0, 4], [0, 10, 1, 10], [-7, -29, -7, 0], [-36, 0, 0, 4], [21, -14, 21, 0], [8, 0, 0, 8], [32, 0, 0, -6], [31, -4, 31, 0], [0, -11, 21, -11], [0, -13, -15, -13], [0, 30, 9, 30], [26, 0, 0, 0], [-4, 0, 0, -32], [0, -7, 41, -7], [-7, 23, -7, 0], [-8, 0, 0, 0], [0, 11, -9, 11], [0, -4, 25, -4], [6, 0, 0, 16], [7, -32, 7, 0], [0, 15, -3, 15], [4, 0, 0, 22], [24, 0, 0, 4], [0, 1, -37, 1], [43, -6, 43, 0], [17, -31, 17, 0], [-15, -21, -15, 0], [-17, -20, -17, 0], [-20, 0, 0, -26], [0, -4, 5, -4], [1, 7, 1, 0], [-16, 0, 0, 26], [0, 3, 25, 3], [-7, -16, -7, 0], [-15, 9, -15, 0], [4, 0, 0, 6], [-18, 0, 0, 4], [-7, -32, -7, 0], [0, 0, 0, -28], [0, 18, 7, 18], [0, 3, 9, 3], [-36, 0, 0, 8], [18, 0, 0, 8], [0, 16, -15, 16], [25, -1, 25, 0], [23, 20, 23, 0], [-5, 14, -5, 0], [-7, -15, -7, 0], [0, -2, 5, -2], [25, -19, 25, 0], [0, -16, 17, -16], [28, 0, 0, 16], [0, -33, 3, -33], [0, -26, -9, -26], [23, -17, 23, 0], [-19, 22, -19, 0], [3, -6, 3, 0], [0, 2, 17, 2], [36, 0, 0, 6], [0, -15, -33, -15], [27, 18, 27, 0], [-32, 0, 0, 4], [0, -16, 37, -16], [-11, -22, -11, 0], [-4, 0, 0, 24], [-9, 12, -9, 0], [0, -27, -25, -27], [0, -21, 19, -21], [-6, 0, 0, 32], [15, 1, 15, 0], [0, 30, -11, 30], [-32, 0, 0, 14], [16, 0, 0, 0], [0, 5, 3, 5], [-7, -7, -7, 0], [0, -2, -21, -2], [47, 11, 47, 0], [-35, -6, -35, 0], [0, 31, 7, 31], [-2, 0, 0, 32], [0, -10, 9, -10], [2, 0, 0, 16], [0, 28, -17, 28], [0, 23, 23, 23], [60, 0, 0, 8], [40, 0, 0, 14], [0, -10, 19, -10], [-23, -25, -23, 0], [0, 17, -15, 17], [0, 7, -51, 7], [25, 12, 25, 0], [23, 9, 23, 0], [-8, 0, 0, -48], [11, 38, 11, 0], [12, 0, 0, 20], [23, -3, 23, 0], [0, 42, 3, 42], [0, 26, 9, 26], [12, 0, 0, -28], [-66, 0, 0, 0], [25, 11, 25, 0], [0, -15, -11, -15], [23, 17, 23, 0], [0, -20, 27, -20], [30, 0, 0, 0], [0, 21, -25, 21], [16, 0, 0, -36], [9, -19, 9, 0], [55, -5, 55, 0], [6, 0, 0, -24], [-48, 0, 0, 14], [0, -8, 33, -8], [31, -13, 31, 0], [12, 0, 0, 18], [-1, 0, -1, 0], [0, 19, 35, 19], [49, -4, 49, 0], [-15, 33, -15, 0], [-56, 0, 0, 10], [0, -10, -21, -10], [9, -1, 9, 0], [52, 0, 0, -12], [5, 34, 5, 0], [-64, 0, 0, 6], [-59, 2, -59, 0], [0, -19, -11, -19], [8, 0, 0, -2], [20, 0, 0, -2], [-36, 0, 0, 28], [27, 2, 27, 0], [0, 18, 31, 18], [55, -5, 55, 0], [13, 18, 13, 0], [0, -13, 15, -13], [-20, 0, 0, -30], [0, -15, 31, -15], [0, -25, -11, -25], [-7, -36, -7, 0], [0, -4, -41, -4], [0, -39, -3, -39], [9, -11, 9, 0], [-57, -17, -57, 0], [40, 0, 0, 0], [-24, 0, 0, -10], [0, -7, 37, -7], [0, 19, -57, 19], [-7, -16, -7, 0], [-16, 0, 0, 2], [7, -31, 7, 0], [36, 0, 0, -30], [43, -22, 43, 0], [-7, -15, -7, 0], [0, 24, 15, 24], [-2, 0, 0, 12], [-32, 0, 0, -26], [5, -14, 5, 0], [0, 9, -7, 9], [16, 0, 0, -6], [0, 1, 19, 1], [10, 0, 0, -12], [0, 12, 17, 12], [0, 6, 41, 6], [-1, -5, -1, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1568_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1568_2_i_o();".
// 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_1568_2_i_o(:prec:=4)
    chi := MakeCharacter_1568_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(1999) - 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_1568_2_i_o();".
// 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_1568_2_i_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1568_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -2, 5, 2, 1]>,<5,R![49, 14, 11, -2, 1]>,<11,R![1, -2, 5, 2, 1]>],Snew);
    return Vf;
end function;