
// Make newform 1360.2.c.f in Magma, downloaded from the LMFDB on 01 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1360_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1360_c_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_1360_2_c_f();".
// 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_1360_2_c_f();".
// 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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [38, 2, -1, -4, 8, -1], [29, 10, -5, -20, 17, -5], [-13, 32, 30, 5, -10, 7], [27, -70, -11, -21, 19, -11], [26, -64, -37, -10, 20, -14]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_1360_c();"
function MakeCharacter_1360_c()
    N := 1360;
    order := 2;
    char_gens := [511, 341, 817, 241];
    v := [2, 2, 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_1360_c_Hecke();"
function MakeCharacter_1360_c_Hecke( : Kf := false)
    N := 1360;
    order := 2;
    char_gens := [511, 341, 817, 241];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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, 0, 0], [0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 2, -1], [0, 0, 0, 1, 1, 0], [1, -1, -1, 0, 0, 0], [0, -2, 1, 2, 0, -1], [0, -2, 0, 0, 0, 0], [0, 0, 0, 4, -4, 1], [0, 0, 0, -4, 0, -2], [0, 0, 0, -3, -1, -2], [0, 0, 0, 2, 2, 2], [0, 0, 0, 2, -4, -2], [-5, -1, -1, 0, 0, 0], [3, 3, -1, 0, 0, 0], [-6, 2, -4, 0, 0, 0], [-2, 0, -2, 0, 0, 0], [0, 0, 0, -4, 6, -6], [-1, -5, 3, 0, 0, 0], [0, 0, 0, -3, 1, 6], [0, 0, 0, 10, 0, -2], [0, 0, 0, 5, -5, 2], [1, 3, 5, 0, 0, 0], [-1, -5, -5, 0, 0, 0], [0, 0, 0, -6, -2, -4], [-3, 5, 1, 0, 0, 0], [5, -1, -7, 0, 0, 0], [0, 0, 0, -2, 6, -3], [0, 0, 0, -6, 0, 4], [0, 0, 0, 4, 6, 0], [1, 1, -3, 0, 0, 0], [0, 0, 0, -3, -5, -4], [9, -1, 3, 0, 0, 0], [0, 0, 0, -11, 5, -4], [2, 2, 8, 0, 0, 0], [-2, 4, 6, 0, 0, 0], [-6, 2, -4, 0, 0, 0], [0, 0, 0, 6, 0, -5], [0, 0, 0, 10, -4, 5], [0, 0, 0, 0, -8, 0], [4, -4, 4, 0, 0, 0], [0, 0, 0, 2, 6, 0], [8, -2, -8, 0, 0, 0], [0, 0, 0, 12, -10, 4], [0, 0, 0, -2, -6, 6], [0, 0, 0, -7, 3, 6], [0, 0, 0, 19, 1, -2], [19, -3, 7, 0, 0, 0], [0, 0, 0, 4, -6, 1], [9, 1, 1, 0, 0, 0], [0, 0, 0, 4, 4, 6], [10, -6, 6, 0, 0, 0], [0, 0, 0, 4, -2, 4], [14, 4, 2, 0, 0, 0], [9, 7, -1, 0, 0, 0], [7, 7, -1, 0, 0, 0], [0, 0, 0, -14, 4, 4], [0, 4, 0, 0, 0, 0], [0, 0, 0, 10, 0, -4], [-14, 2, -12, 0, 0, 0], [0, 0, 0, 8, 2, -3], [8, 2, 2, 0, 0, 0], [-5, -3, -13, 0, 0, 0], [0, 0, 0, 5, -11, 10], [0, 0, 0, -10, 4, -10], [0, 0, 0, 20, -6, -2], [-4, 10, -4, 0, 0, 0], [0, 0, 0, -8, 0, 6], [0, 0, 0, 6, 4, 11], [-2, -14, 8, 0, 0, 0], [14, 8, 0, 0, 0, 0], [2, 6, -2, 0, 0, 0], [0, 0, 0, -4, -2, 7], [5, 11, -9, 0, 0, 0], [0, 0, 0, 1, -1, -4], [-7, 15, -11, 0, 0, 0], [17, 3, -7, 0, 0, 0], [0, 0, 0, 22, 4, 4], [0, 0, 0, -6, -10, 10], [-2, -4, 12, 0, 0, 0], [0, 0, 0, -3, -5, 8], [-3, 11, 1, 0, 0, 0], [0, 0, 0, 11, -7, 4], [-11, 7, -5, 0, 0, 0], [0, 0, 0, 1, 1, -6], [11, -11, 15, 0, 0, 0], [0, 0, 0, -6, -8, 10], [-7, -1, 11, 0, 0, 0], [2, -2, 4, 0, 0, 0], [-15, 3, 5, 0, 0, 0], [15, -9, -1, 0, 0, 0], [0, 0, 0, -13, 5, 0], [0, 0, 0, 20, 10, 3], [-14, 6, -6, 0, 0, 0], [0, 0, 0, -3, 19, -8], [0, 0, 0, -4, 6, -1], [-2, 8, -12, 0, 0, 0], [0, 0, 0, 14, 2, 6], [-7, 5, -7, 0, 0, 0], [0, 0, 0, 8, -4, -14], [0, 0, 0, -12, 8, -1], [-15, 7, 3, 0, 0, 0], [9, 11, -11, 0, 0, 0], [10, 4, 0, 0, 0, 0], [0, 0, 0, 9, -1, -16], [9, -9, 3, 0, 0, 0], [-11, -1, -7, 0, 0, 0], [16, 8, 10, 0, 0, 0], [8, -14, 0, 0, 0, 0], [0, 0, 0, 20, -2, -12], [0, 0, 0, 12, -12, -7], [-8, -10, -2, 0, 0, 0], [0, 0, 0, 20, -2, -4], [0, 0, 0, 5, -19, 4], [-4, -6, -16, 0, 0, 0], [0, 0, 0, -26, 16, -2], [0, 0, 0, -24, 4, 3], [-29, 7, -13, 0, 0, 0], [0, 0, 0, 12, 12, -4], [-6, 2, 18, 0, 0, 0], [-14, 10, -16, 0, 0, 0], [0, 0, 0, 4, -6, 8], [0, 0, 0, 6, -12, 12], [0, 0, 0, 10, -6, 5], [0, 0, 0, 33, 5, 0], [5, 11, 5, 0, 0, 0], [0, 0, 0, -26, 6, -4], [0, 0, 0, 21, 9, -6], [-11, 15, -3, 0, 0, 0], [-4, 6, 18, 0, 0, 0], [-4, -2, 4, 0, 0, 0], [0, 0, 0, -2, 2, -13], [0, 0, 0, -31, -7, 14], [-15, -21, 7, 0, 0, 0], [15, -17, 11, 0, 0, 0], [23, -9, 19, 0, 0, 0], [-3, -1, 11, 0, 0, 0], [0, 0, 0, -4, -14, -7], [-28, 6, -22, 0, 0, 0], [0, 0, 0, -6, 18, -18], [0, 0, 0, -17, 3, -2], [0, 0, 0, 2, -12, 16], [0, 0, 0, -2, 10, 7], [0, 0, 0, 12, -4, -5], [-2, -8, -8, 0, 0, 0], [0, 0, 0, 23, 3, -12], [0, 0, 0, -6, -10, 2], [0, 0, 0, 10, -8, 18], [-6, 6, -10, 0, 0, 0], [-13, -21, 3, 0, 0, 0], [0, 0, 0, -12, 10, 10], [0, 0, 0, 16, -14, -4], [19, -3, -5, 0, 0, 0], [0, 0, 0, -10, -2, -17], [0, 0, 0, -4, 6, 5], [0, 0, 0, -45, 7, -4], [2, 12, -18, 0, 0, 0], [0, 0, 0, 18, -24, 18], [12, -20, -6, 0, 0, 0], [0, 0, 0, -6, 18, -4], [0, 0, 0, -2, -16, 23], [-23, 7, -1, 0, 0, 0], [-9, -5, -17, 0, 0, 0], [20, -8, 8, 0, 0, 0], [-16, 12, 14, 0, 0, 0], [0, 0, 0, 6, 0, 1], [0, 0, 0, 3, 19, 4], [0, 0, 0, -20, -12, 16], [0, 0, 0, -2, 30, 6], [0, 0, 0, -14, 10, 18], [26, 0, -2, 0, 0, 0], [-11, 1, -23, 0, 0, 0], [0, 0, 0, 11, -11, 12], [-31, -1, 3, 0, 0, 0], [8, 10, 16, 0, 0, 0], [0, 0, 0, 0, -16, -12], [0, 0, 0, -13, -13, -18], [0, 0, 0, -16, -8, -10], [15, -7, 19, 0, 0, 0], [1, -5, 9, 0, 0, 0], [-29, -3, -5, 0, 0, 0], [0, 0, 0, 3, -13, 26], [0, 0, 0, 14, 12, 16], [10, -4, 12, 0, 0, 0], [3, -1, 7, 0, 0, 0], [-7, 29, -19, 0, 0, 0], [0, 0, 0, 8, -2, -10], [-21, 11, 3, 0, 0, 0], [0, 0, 0, 20, -30, 16], [0, 0, 0, -27, 19, -14], [0, 0, 0, -12, 16, -2], [0, 0, 0, 6, 2, 17], [-10, 6, 26, 0, 0, 0], [5, -1, -3, 0, 0, 0], [0, 0, 0, -26, -4, 27], [0, 0, 0, -32, -10, 16], [0, 0, 0, -6, -24, 6], [0, 0, 0, 32, -22, 10], [0, 0, 0, 8, 18, 8], [11, 7, 7, 0, 0, 0], [0, 0, 0, 0, -16, -26], [0, 0, 0, -25, 9, -4], [40, -6, -14, 0, 0, 0], [11, 7, -9, 0, 0, 0], [30, 16, 2, 0, 0, 0], [-3, 7, -21, 0, 0, 0], [12, 20, -12, 0, 0, 0], [-33, 13, -1, 0, 0, 0], [0, 0, 0, 4, 0, -16], [20, 26, -4, 0, 0, 0], [0, 0, 0, 0, -16, -18], [5, -21, 17, 0, 0, 0], [0, 0, 0, -6, -2, -1], [35, -1, 15, 0, 0, 0], [0, 0, 0, -35, -9, -2], [0, 0, 0, 10, 12, 10], [7, -3, 7, 0, 0, 0], [6, 16, 0, 0, 0, 0], [0, 0, 0, 6, -10, 7], [-27, -17, -5, 0, 0, 0], [42, 8, 4, 0, 0, 0], [0, 0, 0, 47, -9, 0], [26, 10, 12, 0, 0, 0], [0, 0, 0, -14, 4, 17], [-3, 11, 17, 0, 0, 0], [-3, 15, -23, 0, 0, 0], [0, 0, 0, -12, -12, 6], [0, 0, 0, 9, -7, -14], [-5, -23, 3, 0, 0, 0], [0, 0, 0, -19, -1, 4], [0, -14, -6, 0, 0, 0], [0, 0, 0, -9, -15, 26], [-36, 12, -16, 0, 0, 0], [-17, -15, -5, 0, 0, 0], [-7, -23, 9, 0, 0, 0], [31, 3, -5, 0, 0, 0], [0, 0, 0, -14, 8, -6], [0, 0, 0, 12, -34, 6], [0, 0, 0, 25, -5, -4], [-8, -22, -12, 0, 0, 0], [0, 0, 0, -32, -12, -17], [16, 14, -8, 0, 0, 0], [17, -19, 37, 0, 0, 0], [-35, -3, 1, 0, 0, 0], [0, 0, 0, 12, 12, -22], [0, 0, 0, 41, -15, -6], [0, 0, 0, -28, 8, -3], [0, 0, 0, 21, -9, -8], [-10, -34, 6, 0, 0, 0], [-29, -11, -5, 0, 0, 0], [-2, 6, 28, 0, 0, 0], [0, 0, 0, -10, -14, 18], [-29, -3, -5, 0, 0, 0], [0, 0, 0, -4, 18, 0], [29, -9, -5, 0, 0, 0], [-8, 16, 16, 0, 0, 0], [0, 0, 0, -4, -16, -2], [0, 0, 0, -46, 10, 1], [0, 0, 0, 2, 22, -10], [-15, 3, -33, 0, 0, 0], [0, 0, 0, -2, 4, -3], [-35, 17, -3, 0, 0, 0], [0, 0, 0, -20, -24, 2], [0, 0, 0, -14, -8, -24], [0, 0, 0, 24, -8, 26], [-36, -4, -4, 0, 0, 0], [-10, -32, 4, 0, 0, 0], [-2, -2, -8, 0, 0, 0], [0, 0, 0, 48, 6, 13], [6, -2, 28, 0, 0, 0], [0, 0, 0, 44, -40, 18], [-13, -39, 11, 0, 0, 0], [4, -24, 22, 0, 0, 0], [6, 14, 6, 0, 0, 0], [-16, 0, -26, 0, 0, 0], [43, -15, 39, 0, 0, 0], [5, -27, 21, 0, 0, 0], [37, 1, 9, 0, 0, 0], [7, 1, -29, 0, 0, 0], [-36, 2, 8, 0, 0, 0], [-21, 17, -25, 0, 0, 0], [0, 0, 0, 33, -31, 18], [0, 0, 0, -4, -16, -19], [30, 2, -20, 0, 0, 0], [0, 0, 0, -22, -10, 9], [6, -12, -14, 0, 0, 0], [0, 0, 0, -14, 28, -2], [0, 0, 0, -14, 10, -18], [-8, -6, 20, 0, 0, 0], [27, -27, 35, 0, 0, 0], [0, 0, 0, 0, 30, -34], [0, 0, 0, 28, 0, -9], [-7, 7, 19, 0, 0, 0], [0, 0, 0, 35, -17, -10], [0, 0, 0, 36, 2, -6], [0, 0, 0, 20, -2, -14], [14, 4, -10, 0, 0, 0], [-35, -13, 11, 0, 0, 0], [0, 0, 0, 35, 19, 2], [-13, -15, 7, 0, 0, 0], [-15, -9, 15, 0, 0, 0], [-7, -13, -25, 0, 0, 0], [0, 0, 0, 13, 7, 22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1360_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1360_2_c_f();".
// 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_1360_2_c_f(:prec:=6)
    chi := MakeCharacter_1360_c();
    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_1360_2_c_f();".
// 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_1360_2_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1360_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 0, 16, 0, 8, 0, 1]>,<7,R![100, 0, 104, 0, 28, 0, 1]>],Snew);
    return Vf;
end function;