![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5310, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([58,29,28]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1457, 5310))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5310.1457");
| Modulus: | \(5310\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(885\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(116\) |
![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 \(\chi_{885}(572,\cdot)\) |
![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);
|
\(\chi_{5310}(17,\cdot)\)
\(\chi_{5310}(53,\cdot)\)
\(\chi_{5310}(107,\cdot)\)
\(\chi_{5310}(143,\cdot)\)
\(\chi_{5310}(197,\cdot)\)
\(\chi_{5310}(287,\cdot)\)
\(\chi_{5310}(323,\cdot)\)
\(\chi_{5310}(557,\cdot)\)
\(\chi_{5310}(593,\cdot)\)
\(\chi_{5310}(647,\cdot)\)
\(\chi_{5310}(737,\cdot)\)
\(\chi_{5310}(953,\cdot)\)
\(\chi_{5310}(1007,\cdot)\)
\(\chi_{5310}(1097,\cdot)\)
\(\chi_{5310}(1133,\cdot)\)
\(\chi_{5310}(1187,\cdot)\)
\(\chi_{5310}(1313,\cdot)\)
\(\chi_{5310}(1403,\cdot)\)
\(\chi_{5310}(1457,\cdot)\)
\(\chi_{5310}(1583,\cdot)\)
\(\chi_{5310}(1673,\cdot)\)
\(\chi_{5310}(1727,\cdot)\)
\(\chi_{5310}(1907,\cdot)\)
\(\chi_{5310}(2033,\cdot)\)
\(\chi_{5310}(2087,\cdot)\)
\(\chi_{5310}(2177,\cdot)\)
\(\chi_{5310}(2267,\cdot)\)
\(\chi_{5310}(2447,\cdot)\)
\(\chi_{5310}(2483,\cdot)\)
\(\chi_{5310}(2573,\cdot)\)
...
![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 };
\((1181,3187,3601)\) → \((-1,i,e\left(\frac{7}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5310 }(1457, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{51}{58}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
