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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4307, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([64,29]))
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gp:[g,chi] = znchar(Mod(1998, 4307))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4307.1998");
| Modulus: | \(4307\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4307\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(116\) |
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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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\(\chi_{4307}(27,\cdot)\)
\(\chi_{4307}(46,\cdot)\)
\(\chi_{4307}(100,\cdot)\)
\(\chi_{4307}(192,\cdot)\)
\(\chi_{4307}(265,\cdot)\)
\(\chi_{4307}(411,\cdot)\)
\(\chi_{4307}(484,\cdot)\)
\(\chi_{4307}(538,\cdot)\)
\(\chi_{4307}(557,\cdot)\)
\(\chi_{4307}(611,\cdot)\)
\(\chi_{4307}(684,\cdot)\)
\(\chi_{4307}(757,\cdot)\)
\(\chi_{4307}(776,\cdot)\)
\(\chi_{4307}(830,\cdot)\)
\(\chi_{4307}(995,\cdot)\)
\(\chi_{4307}(1049,\cdot)\)
\(\chi_{4307}(1141,\cdot)\)
\(\chi_{4307}(1195,\cdot)\)
\(\chi_{4307}(1268,\cdot)\)
\(\chi_{4307}(1287,\cdot)\)
\(\chi_{4307}(1360,\cdot)\)
\(\chi_{4307}(1414,\cdot)\)
\(\chi_{4307}(1433,\cdot)\)
\(\chi_{4307}(1487,\cdot)\)
\(\chi_{4307}(1560,\cdot)\)
\(\chi_{4307}(1579,\cdot)\)
\(\chi_{4307}(1779,\cdot)\)
\(\chi_{4307}(1798,\cdot)\)
\(\chi_{4307}(1998,\cdot)\)
\(\chi_{4307}(2090,\cdot)\)
...
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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 };
\((2775,1830)\) → \((e\left(\frac{16}{29}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4307 }(1998, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{13}{116}\right)\) | \(e\left(\frac{63}{116}\right)\) |
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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)
