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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1625, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([13,75]))
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gp:[g,chi] = znchar(Mod(317, 1625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1625.317");
| Modulus: | \(1625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1625\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(100\) |
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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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\(\chi_{1625}(8,\cdot)\)
\(\chi_{1625}(73,\cdot)\)
\(\chi_{1625}(122,\cdot)\)
\(\chi_{1625}(138,\cdot)\)
\(\chi_{1625}(187,\cdot)\)
\(\chi_{1625}(203,\cdot)\)
\(\chi_{1625}(252,\cdot)\)
\(\chi_{1625}(317,\cdot)\)
\(\chi_{1625}(333,\cdot)\)
\(\chi_{1625}(398,\cdot)\)
\(\chi_{1625}(447,\cdot)\)
\(\chi_{1625}(463,\cdot)\)
\(\chi_{1625}(512,\cdot)\)
\(\chi_{1625}(528,\cdot)\)
\(\chi_{1625}(577,\cdot)\)
\(\chi_{1625}(642,\cdot)\)
\(\chi_{1625}(658,\cdot)\)
\(\chi_{1625}(723,\cdot)\)
\(\chi_{1625}(772,\cdot)\)
\(\chi_{1625}(788,\cdot)\)
\(\chi_{1625}(837,\cdot)\)
\(\chi_{1625}(853,\cdot)\)
\(\chi_{1625}(902,\cdot)\)
\(\chi_{1625}(967,\cdot)\)
\(\chi_{1625}(983,\cdot)\)
\(\chi_{1625}(1048,\cdot)\)
\(\chi_{1625}(1097,\cdot)\)
\(\chi_{1625}(1113,\cdot)\)
\(\chi_{1625}(1162,\cdot)\)
\(\chi_{1625}(1178,\cdot)\)
...
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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 };
\((1002,626)\) → \((e\left(\frac{13}{100}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 1625 }(317, a) \) |
\(1\) | \(1\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
