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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2900, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([70,42,5]))
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gp:[g,chi] = znchar(Mod(1539, 2900))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2900.1539");
| Modulus: | \(2900\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2900\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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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\(\chi_{2900}(19,\cdot)\)
\(\chi_{2900}(39,\cdot)\)
\(\chi_{2900}(79,\cdot)\)
\(\chi_{2900}(119,\cdot)\)
\(\chi_{2900}(159,\cdot)\)
\(\chi_{2900}(259,\cdot)\)
\(\chi_{2900}(279,\cdot)\)
\(\chi_{2900}(359,\cdot)\)
\(\chi_{2900}(379,\cdot)\)
\(\chi_{2900}(479,\cdot)\)
\(\chi_{2900}(519,\cdot)\)
\(\chi_{2900}(559,\cdot)\)
\(\chi_{2900}(619,\cdot)\)
\(\chi_{2900}(659,\cdot)\)
\(\chi_{2900}(739,\cdot)\)
\(\chi_{2900}(839,\cdot)\)
\(\chi_{2900}(859,\cdot)\)
\(\chi_{2900}(939,\cdot)\)
\(\chi_{2900}(959,\cdot)\)
\(\chi_{2900}(1059,\cdot)\)
\(\chi_{2900}(1139,\cdot)\)
\(\chi_{2900}(1179,\cdot)\)
\(\chi_{2900}(1239,\cdot)\)
\(\chi_{2900}(1279,\cdot)\)
\(\chi_{2900}(1319,\cdot)\)
\(\chi_{2900}(1419,\cdot)\)
\(\chi_{2900}(1439,\cdot)\)
\(\chi_{2900}(1519,\cdot)\)
\(\chi_{2900}(1539,\cdot)\)
\(\chi_{2900}(1639,\cdot)\)
...
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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 };
\((1451,1277,901)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{1}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2900 }(1539, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{47}{140}\right)\) |
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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)
