![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9400, base_ring=CyclotomicField(230))
M = H._module
chi = DirichletCharacter(H, M([115,0,69,180]))
![Copy content]()
gp:[g,chi] = znchar(Mod(239, 9400))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9400.239");
| Modulus: | \(9400\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4700\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(230\) |
![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: | no, induced from \(\chi_{4700}(239,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{9400}(79,\cdot)\)
\(\chi_{9400}(119,\cdot)\)
\(\chi_{9400}(159,\cdot)\)
\(\chi_{9400}(239,\cdot)\)
\(\chi_{9400}(319,\cdot)\)
\(\chi_{9400}(439,\cdot)\)
\(\chi_{9400}(479,\cdot)\)
\(\chi_{9400}(519,\cdot)\)
\(\chi_{9400}(559,\cdot)\)
\(\chi_{9400}(639,\cdot)\)
\(\chi_{9400}(679,\cdot)\)
\(\chi_{9400}(719,\cdot)\)
\(\chi_{9400}(759,\cdot)\)
\(\chi_{9400}(1239,\cdot)\)
\(\chi_{9400}(1319,\cdot)\)
\(\chi_{9400}(1559,\cdot)\)
\(\chi_{9400}(1679,\cdot)\)
\(\chi_{9400}(1719,\cdot)\)
\(\chi_{9400}(1839,\cdot)\)
\(\chi_{9400}(1959,\cdot)\)
\(\chi_{9400}(2039,\cdot)\)
\(\chi_{9400}(2119,\cdot)\)
\(\chi_{9400}(2319,\cdot)\)
\(\chi_{9400}(2359,\cdot)\)
\(\chi_{9400}(2439,\cdot)\)
\(\chi_{9400}(2519,\cdot)\)
\(\chi_{9400}(2559,\cdot)\)
\(\chi_{9400}(2639,\cdot)\)
\(\chi_{9400}(2879,\cdot)\)
\(\chi_{9400}(3079,\cdot)\)
...
![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 };
\((2351,4701,377,3201)\) → \((-1,1,e\left(\frac{3}{10}\right),e\left(\frac{18}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 9400 }(239, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{115}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{179}{230}\right)\) | \(e\left(\frac{71}{230}\right)\) | \(e\left(\frac{97}{230}\right)\) | \(e\left(\frac{27}{230}\right)\) | \(e\left(\frac{34}{115}\right)\) | \(e\left(\frac{82}{115}\right)\) | \(e\left(\frac{87}{115}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
