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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40051, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([105,14]))
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gp:[g,chi] = znchar(Mod(11604, 40051))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40051.11604");
| Modulus: | \(40051\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(40051\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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);
|
\(\chi_{40051}(21,\cdot)\)
\(\chi_{40051}(230,\cdot)\)
\(\chi_{40051}(857,\cdot)\)
\(\chi_{40051}(1176,\cdot)\)
\(\chi_{40051}(1341,\cdot)\)
\(\chi_{40051}(2342,\cdot)\)
\(\chi_{40051}(3244,\cdot)\)
\(\chi_{40051}(3431,\cdot)\)
\(\chi_{40051}(4333,\cdot)\)
\(\chi_{40051}(5246,\cdot)\)
\(\chi_{40051}(5268,\cdot)\)
\(\chi_{40051}(5587,\cdot)\)
\(\chi_{40051}(6203,\cdot)\)
\(\chi_{40051}(7061,\cdot)\)
\(\chi_{40051}(7479,\cdot)\)
\(\chi_{40051}(7754,\cdot)\)
\(\chi_{40051}(7941,\cdot)\)
\(\chi_{40051}(9008,\cdot)\)
\(\chi_{40051}(9382,\cdot)\)
\(\chi_{40051}(9536,\cdot)\)
\(\chi_{40051}(10548,\cdot)\)
\(\chi_{40051}(10999,\cdot)\)
\(\chi_{40051}(11219,\cdot)\)
\(\chi_{40051}(11274,\cdot)\)
\(\chi_{40051}(11604,\cdot)\)
\(\chi_{40051}(12055,\cdot)\)
\(\chi_{40051}(12660,\cdot)\)
\(\chi_{40051}(12880,\cdot)\)
\(\chi_{40051}(13012,\cdot)\)
\(\chi_{40051}(13133,\cdot)\)
...
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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 };
\((11255,17546)\) → \((e\left(\frac{7}{22}\right),e\left(\frac{7}{165}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 40051 }(11604, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) |
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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)
