![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13380, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([111,0,0,131]))
![Copy content]()
gp:[g,chi] = znchar(Mod(4351, 13380))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13380.4351");
| Modulus: | \(13380\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(892\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(222\) |
![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 \(\chi_{892}(783,\cdot)\) |
![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);
|
\(\chi_{13380}(151,\cdot)\)
\(\chi_{13380}(271,\cdot)\)
\(\chi_{13380}(391,\cdot)\)
\(\chi_{13380}(451,\cdot)\)
\(\chi_{13380}(631,\cdot)\)
\(\chi_{13380}(691,\cdot)\)
\(\chi_{13380}(811,\cdot)\)
\(\chi_{13380}(991,\cdot)\)
\(\chi_{13380}(1291,\cdot)\)
\(\chi_{13380}(1651,\cdot)\)
\(\chi_{13380}(1711,\cdot)\)
\(\chi_{13380}(1891,\cdot)\)
\(\chi_{13380}(2251,\cdot)\)
\(\chi_{13380}(2611,\cdot)\)
\(\chi_{13380}(2911,\cdot)\)
\(\chi_{13380}(3091,\cdot)\)
\(\chi_{13380}(3271,\cdot)\)
\(\chi_{13380}(3391,\cdot)\)
\(\chi_{13380}(3571,\cdot)\)
\(\chi_{13380}(3691,\cdot)\)
\(\chi_{13380}(3811,\cdot)\)
\(\chi_{13380}(3871,\cdot)\)
\(\chi_{13380}(3931,\cdot)\)
\(\chi_{13380}(4111,\cdot)\)
\(\chi_{13380}(4351,\cdot)\)
\(\chi_{13380}(4471,\cdot)\)
\(\chi_{13380}(4531,\cdot)\)
\(\chi_{13380}(4771,\cdot)\)
\(\chi_{13380}(4951,\cdot)\)
\(\chi_{13380}(5071,\cdot)\)
...
![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 };
\((6691,8921,2677,10261)\) → \((-1,1,1,e\left(\frac{131}{222}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 13380 }(4351, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{71}{111}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{221}{222}\right)\) | \(e\left(\frac{76}{111}\right)\) | \(e\left(\frac{59}{111}\right)\) | \(e\left(\frac{197}{222}\right)\) | \(e\left(\frac{77}{111}\right)\) | \(e\left(\frac{14}{37}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
