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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26011, base_ring=CyclotomicField(666))
M = H._module
chi = DirichletCharacter(H, M([296,276]))
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gp:[g,chi] = znchar(Mod(3467, 26011))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26011.3467");
| Modulus: | \(26011\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(26011\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(333\) |
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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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\(\chi_{26011}(63,\cdot)\)
\(\chi_{26011}(137,\cdot)\)
\(\chi_{26011}(195,\cdot)\)
\(\chi_{26011}(454,\cdot)\)
\(\chi_{26011}(491,\cdot)\)
\(\chi_{26011}(655,\cdot)\)
\(\chi_{26011}(766,\cdot)\)
\(\chi_{26011}(840,\cdot)\)
\(\chi_{26011}(898,\cdot)\)
\(\chi_{26011}(1157,\cdot)\)
\(\chi_{26011}(1194,\cdot)\)
\(\chi_{26011}(1358,\cdot)\)
\(\chi_{26011}(1469,\cdot)\)
\(\chi_{26011}(1543,\cdot)\)
\(\chi_{26011}(1601,\cdot)\)
\(\chi_{26011}(1860,\cdot)\)
\(\chi_{26011}(1897,\cdot)\)
\(\chi_{26011}(2061,\cdot)\)
\(\chi_{26011}(2172,\cdot)\)
\(\chi_{26011}(2246,\cdot)\)
\(\chi_{26011}(2304,\cdot)\)
\(\chi_{26011}(2563,\cdot)\)
\(\chi_{26011}(2600,\cdot)\)
\(\chi_{26011}(2764,\cdot)\)
\(\chi_{26011}(2875,\cdot)\)
\(\chi_{26011}(2949,\cdot)\)
\(\chi_{26011}(3007,\cdot)\)
\(\chi_{26011}(3266,\cdot)\)
\(\chi_{26011}(3303,\cdot)\)
\(\chi_{26011}(3467,\cdot)\)
...
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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 };
\((1370,24644)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{46}{111}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 26011 }(3467, a) \) |
\(1\) | \(1\) | \(e\left(\frac{286}{333}\right)\) | \(e\left(\frac{301}{333}\right)\) | \(e\left(\frac{239}{333}\right)\) | \(e\left(\frac{322}{333}\right)\) | \(e\left(\frac{254}{333}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{64}{111}\right)\) | \(e\left(\frac{269}{333}\right)\) | \(e\left(\frac{275}{333}\right)\) | \(e\left(\frac{34}{111}\right)\) |
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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)
