
// Make newform 7360.2.a.cl in Magma, downloaded from the LMFDB on 31 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7360_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_7360_2_a_cl();".
// 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_7360_2_a_cl();".
// 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 := [-16, 18, 8, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-4, 0, 1, 0, 0], [8, 4, -7, -1, 1], [16, -6, -9, 1, 1]];
    Rf_basisdens := [1, 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_7360_a();"
function MakeCharacter_7360_a()
    N := 7360;
    order := 1;
    char_gens := [1151, 5061, 4417, 6721];
    v := [1, 1, 1, 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;

function MakeCharacter_7360_a_Hecke(Kf)
    return MakeCharacter_7360_a();
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, -1, 0], [-1, 0, 0, 0, 0], [0, 0, 1, 1, 1], [0, -1, 0, 0, -2], [-2, 1, -1, 1, 0], [2, -1, 0, -1, 1], [0, 2, 1, -1, -1], [-1, 0, 0, 0, 0], [2, -1, -2, 0, -1], [4, 1, 1, 1, 0], [-2, 0, 0, -1, 3], [-2, 0, 2, 1, -2], [0, 4, -2, 1, -1], [0, 1, 0, -2, 1], [2, 0, 0, -2, -2], [-4, 0, 0, 0, 2], [2, -1, 0, 2, 0], [-8, 2, 2, 1, 1], [-4, -2, 0, 0, 1], [-2, -1, 2, 1, 0], [-4, 0, -2, 3, -3], [-4, -2, 2, 2, 2], [-2, -4, 0, 1, -3], [6, -1, 2, 2, 0], [-6, 0, -4, 2, 0], [-4, -3, -4, -1, 3], [8, -2, 4, -1, -1], [-6, 2, -3, 3, -3], [2, 0, -6, -3, 3], [-8, 3, 2, 0, -1], [0, 3, -2, 5, -2], [2, 0, 1, 4, 0], [0, 1, -4, -3, 4], [-2, 0, -3, 0, -2], [-4, -4, 0, 2, 1], [-2, -2, -2, 7, -3], [-8, -5, 1, 2, -1], [0, 2, -8, 0, -2], [-6, -3, -2, 1, -3], [0, -3, -2, -3, -2], [2, 4, 1, -1, -1], [-8, 0, 0, -4, 6], [-2, 1, 6, 5, 2], [-2, -2, 6, -2, 5], [-4, 2, 0, -1, 5], [4, -6, -2, 4, 0], [-8, 6, 4, 0, -2], [-8, 0, 4, 1, -1], [-2, -6, 6, 1, 1], [6, 1, 2, -3, -2], [0, -5, 0, 1, 4], [2, -6, -2, -4, 4], [4, 2, -3, 2, 4], [14, 3, 2, 1, 0], [-8, -3, -2, -6, -2], [10, -1, 2, -8, -1], [0, -4, -3, -2, 0], [-14, 1, 0, -1, 4], [6, -4, 2, 5, -3], [-4, -4, 2, 0, -2], [10, 6, 0, -4, -2], [-16, 1, -2, 1, -1], [0, 5, 4, -7, -2], [-6, 8, -1, -2, 2], [-6, 8, 3, 0, 6], [-8, 5, 4, 5, 0], [6, 6, -5, -9, -1], [-8, 3, 2, -3, -5], [-6, 5, 2, -2, -3], [-2, 1, 6, -5, 2], [-12, 2, 4, 1, 3], [-8, 4, -4, 2, -2], [2, -4, -6, -4, 4], [-4, -7, 6, -1, -1], [-4, 4, -2, -7, -5], [14, 3, -6, -5, 5], [-18, 0, 2, -4, 1], [-6, -12, 2, 0, 0], [-2, -3, -1, 0, -9], [-4, -6, -6, -1, 7], [-14, 2, 1, -1, -7], [-4, 2, -8, -1, -1], [2, 6, -3, -10, -4], [4, -1, 1, 1, -10], [-16, -1, 3, 2, 3], [-10, 4, -1, -6, 6], [22, 2, 0, 4, 2], [2, 3, 4, 0, 5], [0, 4, -2, -4, 14], [-4, 12, 0, -2, 4], [0, -2, -2, 8, 4], [0, 7, 2, 2, 1], [-16, -5, 6, 5, 0], [8, 3, 8, 7, -6], [-4, -6, -9, 2, 6], [10, -5, -2, 2, 5], [2, -10, -4, -6, -8], [-8, -2, 0, 1, -9], [10, -1, 4, 6, -7], [0, 3, -13, -4, 1], [22, -2, -4, -7, -5], [4, 4, -4, -4, 12], [-6, 0, -4, -8, 8], [4, -5, -6, 1, 5], [6, -5, -8, -5, -2], [-8, -4, 8, -3, 6], [-6, 2, -2, -4, -10], [-8, 13, 4, 1, -1], [-2, -9, -1, -4, 9], [0, -2, -4, 14, -8], [10, -10, 12, -2, 0], [-2, -5, 2, -4, 8], [4, 2, 1, -8, 10], [-4, -8, -6, -7, 1], [-10, 12, 4, -3, -3], [12, -6, 2, -6, 0], [-8, -9, 12, 2, 1], [-2, -1, -5, -11, -6], [0, 8, 0, -4, 0], [-2, -5, -6, -1, -9], [-18, -11, 4, 1, -2], [-6, 8, -12, 0, -8], [8, 5, 5, -6, 9], [-4, 0, 2, -2, 6], [2, -2, 3, 15, 1], [-6, 15, 0, 2, 2], [24, -4, -1, 2, 0], [-12, 3, -4, -1, 1], [18, -6, -4, 4, -6], [24, -7, 8, -5, 8], [-12, 8, 7, 6, 0], [4, -4, -10, -3, -7], [-10, -4, 0, -5, -5], [-2, -9, -5, -12, -7], [10, 0, 0, 4, 12], [-2, 10, 4, -3, 3], [4, -2, 4, 6, -10], [-2, -2, 8, 11, -5], [-10, 9, -4, 3, 5], [8, 1, 2, -5, 0], [10, 2, 8, -12, -2], [0, 3, -12, -4, -1], [-20, 2, -12, 0, -4], [2, -18, -4, 2, 8], [-20, 2, -4, 5, 1], [2, 0, 1, -4, -10], [-18, 1, 8, -1, 0], [-8, 1, 14, 5, 2], [16, -3, 2, -10, -1], [-14, 7, -2, 9, 5], [-14, 4, 2, 8, 6], [0, 4, 1, 10, -14], [-32, -9, 6, 4, -3], [8, 2, -4, 3, 3], [-20, 12, 0, -1, 1], [4, -2, -6, -11, 1], [2, -17, 10, 4, 5], [-2, 0, 3, 7, 5], [22, 1, -2, -5, 5], [0, 3, 3, 2, -5], [-6, 11, -8, -3, -7], [0, -5, -2, -10, -3], [-12, -1, -2, -7, 7], [-2, 4, 9, -7, 3], [-12, 3, -8, -1, -15], [16, 11, 0, -5, -1], [-10, -8, 0, 10, -14], [-10, 8, 12, -11, 7], [-10, 2, -2, 8, -1], [-20, 4, 10, 7, 1], [2, 1, 10, 2, 7], [0, 4, -10, -2, 6], [34, -7, -4, -5, -7], [4, -12, 0, -8, -5], [-26, -12, 6, 5, -3], [-24, 5, 6, 7, 10], [18, -5, 8, 2, -3], [8, -13, 2, 4, 8], [-18, -6, 5, 10, 6], [0, -5, -10, -6, -7], [0, 11, 0, 6, -6], [-2, -5, 11, 11, 4], [18, -2, -4, 16, -16], [-12, -1, 4, -5, -3], [-2, 6, 1, 0, 4], [-38, 2, -9, 0, 0], [24, -2, 10, 7, 3], [-18, 8, 10, 5, 6], [0, -3, 12, -7, 9], [6, 1, 14, 11, 4], [-4, 2, 4, 14, -4], [8, -2, 12, 4, 14], [2, 0, 6, 10, 12], [12, 0, 18, 4, 2], [-2, -2, -9, -13, -1], [30, -2, -6, -3, -5], [-6, 6, 20, 6, -8], [-2, 10, 4, 10, 12], [8, 11, -8, -6, 11], [30, -3, 2, 5, -5], [-8, 6, -7, 10, -14], [-18, 0, -6, -14, 7], [6, -6, 0, 1, -17], [0, 4, 4, -12, -4], [10, -3, 0, 1, 10], [-4, 14, -4, 1, -9], [-8, 16, -8, -3, -4], [6, 9, -3, -16, 15], [-8, 15, -6, -3, -8], [22, -13, 4, -13, -12], [2, -11, -6, -6, 5], [-24, 8, 9, -3, 1], [-4, 2, 2, -6, 20], [20, 12, 0, 10, -6], [22, -12, -6, 5, 1], [-40, -3, -6, -4, 9], [-6, 3, -6, 10, -1], [-20, 10, -3, -4, 12], [18, -7, -16, -7, 2], [-38, 9, -2, 0, -3], [36, -2, -12, 3, -3], [-2, -2, 2, 15, -12], [24, -4, 3, -3, -11], [-28, -1, -10, 0, -7], [10, 7, 2, -2, -7], [14, -16, -7, 3, 5], [4, 5, 5, 15, 10], [12, -6, -3, 0, -8], [-12, 4, -4, 4, -16], [-22, 16, 9, -4, -10], [16, -7, 12, -4, 0], [0, -6, 16, -6, 12], [-2, -7, -12, 9, -7], [-4, -8, -6, 6, -10], [-4, 4, -6, -3, -9], [18, 12, 0, 12, -10], [22, -8, 2, -3, 17], [8, -13, -2, -11, -16], [12, -12, 4, 16, 9], [-20, 12, -2, 0, 4], [24, -19, 2, 1, 12], [-8, -7, 4, 26, 7], [-6, 13, -4, -12, -5], [2, -10, 16, 4, 0], [12, -13, 7, -5, -4], [24, 11, 8, 12, -7], [20, -4, -1, -4, 8], [4, -9, -10, 3, 1], [-4, -3, 12, 7, -3], [-14, -22, 6, 4, -2], [-14, 12, 10, -6, 2], [-4, 2, 9, 18, 2], [22, 0, -4, -19, -3], [6, -2, -8, -4, 2], [24, -13, 12, -7, -2], [-30, 13, 0, -4, 10], [-36, 12, 0, -6, -8], [22, -3, -13, -7, 4], [22, 7, 12, 3, -2], [-12, -5, 4, -1, -13], [8, 2, 8, -3, 15], [18, -4, 6, 12, -6], [-26, -6, -2, 7, 1], [-22, -6, -4, 8, 6], [16, -9, -8, 14, -12], [18, -9, 0, -6, -4], [-10, -6, -10, 3, 11], [-16, -12, -14, -5, 15], [-18, -11, -15, -13, 6], [-30, -15, 2, -10, 3], [4, 14, 0, -12, -6], [6, 10, 9, -7, -3], [12, -10, 14, -7, 1], [22, 1, -4, -13, 4], [8, 5, 10, 4, 5], [8, 7, 14, 11, 3], [-46, 4, -8, 0, 2], [18, 8, -8, -8, -4], [-20, 6, 3, 0, 10], [-24, -7, -14, -16, 3], [8, -2, 0, 18, 4], [-16, 15, 6, -2, 6], [2, 16, 0, -14, -10], [-24, 2, 1, 2, 0], [-8, 9, -8, 17, -3], [-10, -4, -16, 4, 0], [22, -2, 12, 23, 7], [-24, -17, 4, 25, -11], [22, -7, 0, 1, -9], [26, 8, 8, -18, 6], [4, 6, -8, -22, 2], [-22, 4, 14, -4, 4], [0, 6, 3, -9, 3], [-2, -6, -10, -6, 11], [30, 11, -10, -7, 5], [0, 4, -18, 12, -12], [-2, -18, 14, 0, -7], [-8, 13, 2, -15, 12], [0, -2, -8, -7, 16], [14, -2, 9, 7, 7], [-2, -8, -2, -11, 13], [16, -6, -4, 0, 8], [-28, -13, 2, 12, -3], [4, -22, -7, -8, -2], [6, 19, -10, -3, 4], [-24, 15, 2, -5, -5], [-10, -6, -3, 0, 4], [0, -6, 10, 6, 4], [2, -3, 30, 9, 5], [0, 9, 4, 0, -1], [-14, 24, 5, -13, -1], [2, 18, -14, -6, -4], [16, 4, 8, -3, 22], [-8, 18, -17, -1, -3], [6, 12, 6, -3, -15], [16, -3, -6, -11, 12], [-4, -15, -1, -5, -2], [30, 14, 0, 8, -6], [30, -4, 18, -13, 4], [24, 3, -20, -8, 4], [22, -1, 18, 16, 10], [18, -9, 6, -4, 25], [16, -18, 2, -4, -6], [10, -3, 0, 15, -1], [2, -16, -6, 4, -10], [-20, 8, 3, -6, 6], [0, 3, -15, 6, -1], [24, 6, -19, -15, 7], [10, 12, 20, 6, -12], [2, -1, -24, -6, -9], [-10, -4, 10, 16, -1], [56, 7, 4, 9, 0], [-24, -11, -11, -12, 7], [-20, 0, 2, 3, -27], [40, 5, 1, 16, -13], [-18, 11, 2, 1, -27], [26, 0, -10, 5, -9], [14, 10, -2, 1, 2], [0, 3, -2, -8, 14], [-22, 9, -8, -15, -10], [22, -11, 2, 2, -8], [26, -20, 10, -4, 10], [8, -2, -18, -10, 4], [-30, 4, 8, -2, 10], [-12, -20, 4, 4, -20], [-14, 1, -2, -22, -7], [-32, 4, -13, -4, -20], [-40, -12, -14, -16, 14], [10, -6, 20, 16, 4], [-16, 7, 10, -3, 12], [-34, 13, 8, -15, 2], [10, -12, -2, 12, -4], [4, -19, -4, -11, 3], [-6, -18, 8, -8, -2], [-14, -10, -12, -8, -4], [-8, 6, 12, 10, -18], [-4, 8, -9, 14, -18], [-10, -5, 0, -3, 8], [0, -24, 10, -4, -6], [50, 4, -6, -10, -2], [10, 25, 10, -8, -9], [8, 3, 6, 8, -5], [-12, -1, 2, 9, -21], [8, 2, -20, -19, -10], [14, 11, -16, -1, -2], [-34, 14, -6, 12, 15], [20, 14, 6, 11, -17], [10, 30, -4, -2, 2], [16, 3, -6, -7, 2], [-4, 10, -16, 22, -14], [-48, 1, 0, 12, 5], [-18, 3, -18, -21, -7], [40, 16, -12, 2, -2], [-34, -5, 13, -11, 14], [-8, 3, -6, -19, 28], [-28, -18, -2, -9, -7], [-38, 11, -20, -5, -5], [-30, 0, 0, 8, 2], [-26, -5, 10, 27, 4], [30, -7, -2, -7, -7], [-26, -13, 6, 6, -10], [-32, 12, -4, 16, 14], [6, -7, -16, 1, -6], [20, -2, -14, -7, -3], [32, -15, 18, -2, 1], [-20, 1, -3, 1, 12], [10, 4, 19, 18, 12], [20, 0, 10, 8, -6], [20, -8, 15, 12, 14], [10, 10, -8, -10, 28], [-14, 26, -7, 4, -20], [-16, 15, 4, -1, -2], [28, 2, 0, -20, 2], [20, 0, -20, -11, -11], [10, 10, 21, 10, -4], [-4, -12, 4, 15, -5], [-2, 4, -4, 5, 25], [-52, -6, -15, 0, 14], [18, -18, -12, 6, 4], [26, -1, 8, 10, -5], [30, -15, -10, 13, -18], [-36, 12, -10, -1, -7], [2, -12, -13, 20, 0], [-14, 9, -8, -22, 7], [52, -6, 20, -12, 3], [14, -16, -14, 1, -7], [-18, -3, 23, 6, 5], [8, 12, 0, -9, -15], [28, -12, 0, -4, 16], [-2, -1, 10, 5, -8], [-2, 4, 6, 18, 7], [24, -10, 2, -19, -15], [28, 9, 14, 29, 3], [22, 16, -5, 21, -13], [26, 1, 18, 8, -17], [16, 24, -17, -8, 2], [40, 10, -2, -6, 0], [14, -3, 6, 10, 16], [8, 8, 1, -21, 7], [-6, -9, -12, -4, -4], [14, 2, -22, 5, -17], [-16, 10, 6, -16, 2], [-36, 12, -6, 12, 0], [6, -21, 0, 23, 0], [26, 1, 4, -25, 4], [0, -10, -14, 17, 5], [-10, -6, -11, 14, -16], [8, -11, 12, 23, 10], [-12, 16, -4, 22, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7360_2_a_cl();".
// 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_7360_2_a_cl(:prec:=5)
    chi := MakeCharacter_7360_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_7360_2_a_cl();".
// 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_7360_2_a_cl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 15, -7, -10, 1, 1]>,<7,R![-256, 130, 52, -29, -1, 1]>,<11,R![592, 626, -100, -49, 3, 1]>,<13,R![74, -249, -179, -16, 7, 1]>,<17,R![16, -48, 34, 5, -9, 1]>],Snew);
    return Vf;
end function;