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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10309, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([105,117]))
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gp:[g,chi] = znchar(Mod(1871, 10309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10309.1871");
| Modulus: | \(10309\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10309\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(130\) |
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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);
|
\(\chi_{10309}(64,\cdot)\)
\(\chi_{10309}(285,\cdot)\)
\(\chi_{10309}(454,\cdot)\)
\(\chi_{10309}(662,\cdot)\)
\(\chi_{10309}(857,\cdot)\)
\(\chi_{10309}(1078,\cdot)\)
\(\chi_{10309}(1247,\cdot)\)
\(\chi_{10309}(1455,\cdot)\)
\(\chi_{10309}(1650,\cdot)\)
\(\chi_{10309}(1871,\cdot)\)
\(\chi_{10309}(2040,\cdot)\)
\(\chi_{10309}(2248,\cdot)\)
\(\chi_{10309}(2443,\cdot)\)
\(\chi_{10309}(2664,\cdot)\)
\(\chi_{10309}(2833,\cdot)\)
\(\chi_{10309}(3236,\cdot)\)
\(\chi_{10309}(3457,\cdot)\)
\(\chi_{10309}(3626,\cdot)\)
\(\chi_{10309}(3834,\cdot)\)
\(\chi_{10309}(4029,\cdot)\)
\(\chi_{10309}(4250,\cdot)\)
\(\chi_{10309}(4419,\cdot)\)
\(\chi_{10309}(4627,\cdot)\)
\(\chi_{10309}(4822,\cdot)\)
\(\chi_{10309}(5043,\cdot)\)
\(\chi_{10309}(5212,\cdot)\)
\(\chi_{10309}(5420,\cdot)\)
\(\chi_{10309}(5615,\cdot)\)
\(\chi_{10309}(5836,\cdot)\)
\(\chi_{10309}(6005,\cdot)\)
...
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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 };
\((8114,2198)\) → \((e\left(\frac{21}{26}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 10309 }(1871, a) \) |
\(1\) | \(1\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{9}{13}\right)\) |
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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)
