![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78585, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([390,585,295,312]))
![Copy content]()
gp:[g,chi] = znchar(Mod(10238, 78585))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78585.10238");
| Modulus: | \(78585\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(78585\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(780\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{78585}(2,\cdot)\)
\(\chi_{78585}(128,\cdot)\)
\(\chi_{78585}(977,\cdot)\)
\(\chi_{78585}(1397,\cdot)\)
\(\chi_{78585}(2048,\cdot)\)
\(\chi_{78585}(2147,\cdot)\)
\(\chi_{78585}(2372,\cdot)\)
\(\chi_{78585}(2732,\cdot)\)
\(\chi_{78585}(3443,\cdot)\)
\(\chi_{78585}(3542,\cdot)\)
\(\chi_{78585}(4127,\cdot)\)
\(\chi_{78585}(4193,\cdot)\)
\(\chi_{78585}(4418,\cdot)\)
\(\chi_{78585}(4778,\cdot)\)
\(\chi_{78585}(5588,\cdot)\)
\(\chi_{78585}(6047,\cdot)\)
\(\chi_{78585}(7022,\cdot)\)
\(\chi_{78585}(7442,\cdot)\)
\(\chi_{78585}(8417,\cdot)\)
\(\chi_{78585}(8777,\cdot)\)
\(\chi_{78585}(9068,\cdot)\)
\(\chi_{78585}(9488,\cdot)\)
\(\chi_{78585}(9587,\cdot)\)
\(\chi_{78585}(10172,\cdot)\)
\(\chi_{78585}(10238,\cdot)\)
\(\chi_{78585}(10463,\cdot)\)
\(\chi_{78585}(10823,\cdot)\)
\(\chi_{78585}(11633,\cdot)\)
\(\chi_{78585}(12092,\cdot)\)
\(\chi_{78585}(12218,\cdot)\)
...
![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 };
\((52391,47152,1861,32956)\) → \((-1,-i,e\left(\frac{59}{156}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 78585 }(10238, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{89}{390}\right)\) | \(e\left(\frac{89}{195}\right)\) | \(e\left(\frac{163}{390}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{511}{780}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{178}{195}\right)\) | \(e\left(\frac{209}{780}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
