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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257725, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([78,450,533]))
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gp:[g,chi] = znchar(Mod(21254, 257725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("257725.21254");
| Modulus: | \(257725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(257725\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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: | odd |
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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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\(\chi_{257725}(519,\cdot)\)
\(\chi_{257725}(714,\cdot)\)
\(\chi_{257725}(909,\cdot)\)
\(\chi_{257725}(1429,\cdot)\)
\(\chi_{257725}(1494,\cdot)\)
\(\chi_{257725}(4094,\cdot)\)
\(\chi_{257725}(5004,\cdot)\)
\(\chi_{257725}(5264,\cdot)\)
\(\chi_{257725}(8904,\cdot)\)
\(\chi_{257725}(9034,\cdot)\)
\(\chi_{257725}(11634,\cdot)\)
\(\chi_{257725}(12479,\cdot)\)
\(\chi_{257725}(12739,\cdot)\)
\(\chi_{257725}(13064,\cdot)\)
\(\chi_{257725}(18134,\cdot)\)
\(\chi_{257725}(20344,\cdot)\)
\(\chi_{257725}(20539,\cdot)\)
\(\chi_{257725}(20734,\cdot)\)
\(\chi_{257725}(21254,\cdot)\)
\(\chi_{257725}(21319,\cdot)\)
\(\chi_{257725}(23919,\cdot)\)
\(\chi_{257725}(24829,\cdot)\)
\(\chi_{257725}(25089,\cdot)\)
\(\chi_{257725}(28859,\cdot)\)
\(\chi_{257725}(31459,\cdot)\)
\(\chi_{257725}(32304,\cdot)\)
\(\chi_{257725}(32564,\cdot)\)
\(\chi_{257725}(32889,\cdot)\)
\(\chi_{257725}(37569,\cdot)\)
\(\chi_{257725}(37959,\cdot)\)
...
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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 };
\((144327,193676,177451)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{15}{26}\right),e\left(\frac{41}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 257725 }(21254, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{281}{780}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{281}{390}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{557}{780}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{71}{260}\right)\) | \(e\left(\frac{23}{390}\right)\) | \(e\left(\frac{29}{390}\right)\) |
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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)
