![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3485, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([60,45,44]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1153, 3485))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3485.1153");
| Modulus: | \(3485\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(3485\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(80\) |
![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);
|
\(\chi_{3485}(197,\cdot)\)
\(\chi_{3485}(282,\cdot)\)
\(\chi_{3485}(402,\cdot)\)
\(\chi_{3485}(487,\cdot)\)
\(\chi_{3485}(513,\cdot)\)
\(\chi_{3485}(743,\cdot)\)
\(\chi_{3485}(822,\cdot)\)
\(\chi_{3485}(828,\cdot)\)
\(\chi_{3485}(923,\cdot)\)
\(\chi_{3485}(1023,\cdot)\)
\(\chi_{3485}(1027,\cdot)\)
\(\chi_{3485}(1153,\cdot)\)
\(\chi_{3485}(1238,\cdot)\)
\(\chi_{3485}(1332,\cdot)\)
\(\chi_{3485}(1433,\cdot)\)
\(\chi_{3485}(1537,\cdot)\)
\(\chi_{3485}(1642,\cdot)\)
\(\chi_{3485}(1847,\cdot)\)
\(\chi_{3485}(2152,\cdot)\)
\(\chi_{3485}(2357,\cdot)\)
\(\chi_{3485}(2358,\cdot)\)
\(\chi_{3485}(2383,\cdot)\)
\(\chi_{3485}(2468,\cdot)\)
\(\chi_{3485}(2768,\cdot)\)
\(\chi_{3485}(2793,\cdot)\)
\(\chi_{3485}(2862,\cdot)\)
\(\chi_{3485}(2868,\cdot)\)
\(\chi_{3485}(2878,\cdot)\)
\(\chi_{3485}(2947,\cdot)\)
\(\chi_{3485}(3067,\cdot)\)
...
![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 };
\((2092,411,2466)\) → \((-i,e\left(\frac{9}{16}\right),e\left(\frac{11}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3485 }(1153, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
