![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,33,25]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1239, 2300))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2300.1239");
| Modulus: | \(2300\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2300\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(110\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
gp:charorder(g,chi)
![Copy content]()
magma:Order(chi);
|
| Real: | no |
![Copy content]()
sage:chi.multiplicative_order() <= 2
![Copy content]()
gp:charorder(g,chi) <= 2
![Copy content]()
magma:Order(chi) le 2;
|
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2300}(19,\cdot)\)
\(\chi_{2300}(79,\cdot)\)
\(\chi_{2300}(159,\cdot)\)
\(\chi_{2300}(319,\cdot)\)
\(\chi_{2300}(339,\cdot)\)
\(\chi_{2300}(359,\cdot)\)
\(\chi_{2300}(379,\cdot)\)
\(\chi_{2300}(419,\cdot)\)
\(\chi_{2300}(479,\cdot)\)
\(\chi_{2300}(539,\cdot)\)
\(\chi_{2300}(559,\cdot)\)
\(\chi_{2300}(619,\cdot)\)
\(\chi_{2300}(659,\cdot)\)
\(\chi_{2300}(779,\cdot)\)
\(\chi_{2300}(819,\cdot)\)
\(\chi_{2300}(839,\cdot)\)
\(\chi_{2300}(879,\cdot)\)
\(\chi_{2300}(939,\cdot)\)
\(\chi_{2300}(1019,\cdot)\)
\(\chi_{2300}(1079,\cdot)\)
\(\chi_{2300}(1119,\cdot)\)
\(\chi_{2300}(1239,\cdot)\)
\(\chi_{2300}(1259,\cdot)\)
\(\chi_{2300}(1279,\cdot)\)
\(\chi_{2300}(1339,\cdot)\)
\(\chi_{2300}(1459,\cdot)\)
\(\chi_{2300}(1479,\cdot)\)
\(\chi_{2300}(1539,\cdot)\)
\(\chi_{2300}(1579,\cdot)\)
\(\chi_{2300}(1719,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
![Copy content]()
magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
| Field of values: |
$\Q(\zeta_{55})$ |
![Copy content]()
sage:CyclotomicField(chi.multiplicative_order())
![Copy content]()
gp:nfinit(polcyclo(charorder(g,chi)))
![Copy content]()
magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 110 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((1151,277,1201)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 2300 }(1239, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) |
![Copy content]()
sage:chi(x) # x integer
![Copy content]()
gp:chareval(g,chi,x) \\ x integer, value in Q/Z
![Copy content]()
magma:chi(x)
