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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5921, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([171,180]))
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gp:[g,chi] = znchar(Mod(2472, 5921))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5921.2472");
| Modulus: | \(5921\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5921\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(190\) |
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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);
|
\(\chi_{5921}(153,\cdot)\)
\(\chi_{5921}(368,\cdot)\)
\(\chi_{5921}(387,\cdot)\)
\(\chi_{5921}(418,\cdot)\)
\(\chi_{5921}(542,\cdot)\)
\(\chi_{5921}(680,\cdot)\)
\(\chi_{5921}(709,\cdot)\)
\(\chi_{5921}(833,\cdot)\)
\(\chi_{5921}(914,\cdot)\)
\(\chi_{5921}(1007,\cdot)\)
\(\chi_{5921}(1108,\cdot)\)
\(\chi_{5921}(1176,\cdot)\)
\(\chi_{5921}(1267,\cdot)\)
\(\chi_{5921}(1300,\cdot)\)
\(\chi_{5921}(1362,\cdot)\)
\(\chi_{5921}(1517,\cdot)\)
\(\chi_{5921}(1534,\cdot)\)
\(\chi_{5921}(1635,\cdot)\)
\(\chi_{5921}(1751,\cdot)\)
\(\chi_{5921}(1844,\cdot)\)
\(\chi_{5921}(2106,\cdot)\)
\(\chi_{5921}(2131,\cdot)\)
\(\chi_{5921}(2137,\cdot)\)
\(\chi_{5921}(2255,\cdot)\)
\(\chi_{5921}(2261,\cdot)\)
\(\chi_{5921}(2278,\cdot)\)
\(\chi_{5921}(2317,\cdot)\)
\(\chi_{5921}(2445,\cdot)\)
\(\chi_{5921}(2472,\cdot)\)
\(\chi_{5921}(2619,\cdot)\)
...
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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 };
\((3630,2884)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{18}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5921 }(2472, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{27}{95}\right)\) | \(e\left(\frac{151}{190}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{81}{95}\right)\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{43}{190}\right)\) |
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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)
