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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19951, base_ring=CyclotomicField(280))
M = H._module
chi = DirichletCharacter(H, M([4,149]))
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gp:[g,chi] = znchar(Mod(4764, 19951))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19951.4764");
| Modulus: | \(19951\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19951\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(280\) |
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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);
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| 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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\(\chi_{19951}(447,\cdot)\)
\(\chi_{19951}(575,\cdot)\)
\(\chi_{19951}(636,\cdot)\)
\(\chi_{19951}(772,\cdot)\)
\(\chi_{19951}(914,\cdot)\)
\(\chi_{19951}(1218,\cdot)\)
\(\chi_{19951}(1417,\cdot)\)
\(\chi_{19951}(1604,\cdot)\)
\(\chi_{19951}(1631,\cdot)\)
\(\chi_{19951}(1806,\cdot)\)
\(\chi_{19951}(1817,\cdot)\)
\(\chi_{19951}(2550,\cdot)\)
\(\chi_{19951}(2836,\cdot)\)
\(\chi_{19951}(3477,\cdot)\)
\(\chi_{19951}(3548,\cdot)\)
\(\chi_{19951}(3890,\cdot)\)
\(\chi_{19951}(4009,\cdot)\)
\(\chi_{19951}(4312,\cdot)\)
\(\chi_{19951}(4392,\cdot)\)
\(\chi_{19951}(4469,\cdot)\)
\(\chi_{19951}(4603,\cdot)\)
\(\chi_{19951}(4606,\cdot)\)
\(\chi_{19951}(4764,\cdot)\)
\(\chi_{19951}(5309,\cdot)\)
\(\chi_{19951}(5394,\cdot)\)
\(\chi_{19951}(5952,\cdot)\)
\(\chi_{19951}(6134,\cdot)\)
\(\chi_{19951}(6171,\cdot)\)
\(\chi_{19951}(6269,\cdot)\)
\(\chi_{19951}(6505,\cdot)\)
...
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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 };
\((19390,13491)\) → \((e\left(\frac{1}{70}\right),e\left(\frac{149}{280}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 19951 }(4764, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{253}{280}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{153}{280}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{3}{40}\right)\) |
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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)
