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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([663,170,78]))
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gp:[g,chi] = znchar(Mod(13972, 257725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("257725.13972");
| 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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\(\chi_{257725}(1772,\cdot)\)
\(\chi_{257725}(3163,\cdot)\)
\(\chi_{257725}(3748,\cdot)\)
\(\chi_{257725}(5958,\cdot)\)
\(\chi_{257725}(6127,\cdot)\)
\(\chi_{257725}(9812,\cdot)\)
\(\chi_{257725}(13972,\cdot)\)
\(\chi_{257725}(15253,\cdot)\)
\(\chi_{257725}(15363,\cdot)\)
\(\chi_{257725}(15948,\cdot)\)
\(\chi_{257725}(16267,\cdot)\)
\(\chi_{257725}(17437,\cdot)\)
\(\chi_{257725}(18158,\cdot)\)
\(\chi_{257725}(18327,\cdot)\)
\(\chi_{257725}(21597,\cdot)\)
\(\chi_{257725}(22988,\cdot)\)
\(\chi_{257725}(23573,\cdot)\)
\(\chi_{257725}(25783,\cdot)\)
\(\chi_{257725}(25952,\cdot)\)
\(\chi_{257725}(27453,\cdot)\)
\(\chi_{257725}(28467,\cdot)\)
\(\chi_{257725}(29637,\cdot)\)
\(\chi_{257725}(33797,\cdot)\)
\(\chi_{257725}(35078,\cdot)\)
\(\chi_{257725}(35188,\cdot)\)
\(\chi_{257725}(35773,\cdot)\)
\(\chi_{257725}(36092,\cdot)\)
\(\chi_{257725}(37262,\cdot)\)
\(\chi_{257725}(37983,\cdot)\)
\(\chi_{257725}(38152,\cdot)\)
...
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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 };
\((144327,193676,177451)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{17}{78}\right),e\left(\frac{1}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 257725 }(13972, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{131}{780}\right)\) | \(e\left(\frac{449}{780}\right)\) | \(e\left(\frac{131}{390}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{367}{780}\right)\) | \(e\left(\frac{131}{260}\right)\) | \(e\left(\frac{59}{390}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{83}{130}\right)\) |
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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)
