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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(92455, base_ring=CyclotomicField(820))
M = H._module
chi = DirichletCharacter(H, M([410,328,639]))
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gp:[g,chi] = znchar(Mod(2304, 92455))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("92455.2304");
| Modulus: | \(92455\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(92455\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(820\) |
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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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\(\chi_{92455}(49,\cdot)\)
\(\chi_{92455}(289,\cdot)\)
\(\chi_{92455}(389,\cdot)\)
\(\chi_{92455}(554,\cdot)\)
\(\chi_{92455}(1604,\cdot)\)
\(\chi_{92455}(1884,\cdot)\)
\(\chi_{92455}(1919,\cdot)\)
\(\chi_{92455}(2209,\cdot)\)
\(\chi_{92455}(2304,\cdot)\)
\(\chi_{92455}(2544,\cdot)\)
\(\chi_{92455}(2644,\cdot)\)
\(\chi_{92455}(2809,\cdot)\)
\(\chi_{92455}(3859,\cdot)\)
\(\chi_{92455}(4139,\cdot)\)
\(\chi_{92455}(4174,\cdot)\)
\(\chi_{92455}(4464,\cdot)\)
\(\chi_{92455}(4559,\cdot)\)
\(\chi_{92455}(4799,\cdot)\)
\(\chi_{92455}(4899,\cdot)\)
\(\chi_{92455}(5064,\cdot)\)
\(\chi_{92455}(6114,\cdot)\)
\(\chi_{92455}(6394,\cdot)\)
\(\chi_{92455}(6429,\cdot)\)
\(\chi_{92455}(6719,\cdot)\)
\(\chi_{92455}(6814,\cdot)\)
\(\chi_{92455}(7054,\cdot)\)
\(\chi_{92455}(7154,\cdot)\)
\(\chi_{92455}(7319,\cdot)\)
\(\chi_{92455}(8369,\cdot)\)
\(\chi_{92455}(8649,\cdot)\)
...
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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 };
\((18492,8406,50436)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{639}{820}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 92455 }(2304, a) \) |
\(1\) | \(1\) | \(e\left(\frac{128}{205}\right)\) | \(e\left(\frac{619}{820}\right)\) | \(e\left(\frac{51}{205}\right)\) | \(e\left(\frac{311}{820}\right)\) | \(e\left(\frac{747}{820}\right)\) | \(e\left(\frac{179}{205}\right)\) | \(e\left(\frac{209}{410}\right)\) | \(e\left(\frac{3}{820}\right)\) | \(e\left(\frac{307}{820}\right)\) | \(e\left(\frac{439}{820}\right)\) |
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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)
