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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,54]))
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gp:[g,chi] = znchar(Mod(181, 1210))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1210.181");
| Modulus: | \(1210\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(121\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(55\) |
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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: | no, induced from \(\chi_{121}(60,\cdot)\) |
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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);
|
\(\chi_{1210}(31,\cdot)\)
\(\chi_{1210}(71,\cdot)\)
\(\chi_{1210}(91,\cdot)\)
\(\chi_{1210}(141,\cdot)\)
\(\chi_{1210}(181,\cdot)\)
\(\chi_{1210}(191,\cdot)\)
\(\chi_{1210}(201,\cdot)\)
\(\chi_{1210}(291,\cdot)\)
\(\chi_{1210}(301,\cdot)\)
\(\chi_{1210}(311,\cdot)\)
\(\chi_{1210}(361,\cdot)\)
\(\chi_{1210}(401,\cdot)\)
\(\chi_{1210}(411,\cdot)\)
\(\chi_{1210}(421,\cdot)\)
\(\chi_{1210}(471,\cdot)\)
\(\chi_{1210}(521,\cdot)\)
\(\chi_{1210}(531,\cdot)\)
\(\chi_{1210}(581,\cdot)\)
\(\chi_{1210}(621,\cdot)\)
\(\chi_{1210}(631,\cdot)\)
\(\chi_{1210}(641,\cdot)\)
\(\chi_{1210}(691,\cdot)\)
\(\chi_{1210}(731,\cdot)\)
\(\chi_{1210}(741,\cdot)\)
\(\chi_{1210}(751,\cdot)\)
\(\chi_{1210}(801,\cdot)\)
\(\chi_{1210}(841,\cdot)\)
\(\chi_{1210}(851,\cdot)\)
\(\chi_{1210}(861,\cdot)\)
\(\chi_{1210}(911,\cdot)\)
...
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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 };
\((727,1091)\) → \((1,e\left(\frac{27}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1210 }(181, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{55}\right)\) |
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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)
