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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4335, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([34,51,66]))
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gp:[g,chi] = znchar(Mod(3263, 4335))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4335.3263");
| Modulus: | \(4335\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4335\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(68\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{4335}(152,\cdot)\)
\(\chi_{4335}(203,\cdot)\)
\(\chi_{4335}(407,\cdot)\)
\(\chi_{4335}(458,\cdot)\)
\(\chi_{4335}(662,\cdot)\)
\(\chi_{4335}(713,\cdot)\)
\(\chi_{4335}(917,\cdot)\)
\(\chi_{4335}(968,\cdot)\)
\(\chi_{4335}(1172,\cdot)\)
\(\chi_{4335}(1223,\cdot)\)
\(\chi_{4335}(1427,\cdot)\)
\(\chi_{4335}(1478,\cdot)\)
\(\chi_{4335}(1682,\cdot)\)
\(\chi_{4335}(1937,\cdot)\)
\(\chi_{4335}(1988,\cdot)\)
\(\chi_{4335}(2192,\cdot)\)
\(\chi_{4335}(2243,\cdot)\)
\(\chi_{4335}(2447,\cdot)\)
\(\chi_{4335}(2498,\cdot)\)
\(\chi_{4335}(2702,\cdot)\)
\(\chi_{4335}(2753,\cdot)\)
\(\chi_{4335}(2957,\cdot)\)
\(\chi_{4335}(3008,\cdot)\)
\(\chi_{4335}(3212,\cdot)\)
\(\chi_{4335}(3263,\cdot)\)
\(\chi_{4335}(3518,\cdot)\)
\(\chi_{4335}(3722,\cdot)\)
\(\chi_{4335}(3773,\cdot)\)
\(\chi_{4335}(3977,\cdot)\)
\(\chi_{4335}(4028,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((2891,2602,2026)\) → \((-1,-i,e\left(\frac{33}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 4335 }(3263, a) \) |
\(1\) | \(1\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{33}{68}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
