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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1832, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([0,114,155]))
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gp:[g,chi] = znchar(Mod(381, 1832))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1832.381");
| Modulus: | \(1832\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1832\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(228\) |
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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);
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\(\chi_{1832}(29,\cdot)\)
\(\chi_{1832}(69,\cdot)\)
\(\chi_{1832}(77,\cdot)\)
\(\chi_{1832}(117,\cdot)\)
\(\chi_{1832}(133,\cdot)\)
\(\chi_{1832}(157,\cdot)\)
\(\chi_{1832}(189,\cdot)\)
\(\chi_{1832}(205,\cdot)\)
\(\chi_{1832}(253,\cdot)\)
\(\chi_{1832}(269,\cdot)\)
\(\chi_{1832}(301,\cdot)\)
\(\chi_{1832}(325,\cdot)\)
\(\chi_{1832}(341,\cdot)\)
\(\chi_{1832}(381,\cdot)\)
\(\chi_{1832}(389,\cdot)\)
\(\chi_{1832}(429,\cdot)\)
\(\chi_{1832}(493,\cdot)\)
\(\chi_{1832}(517,\cdot)\)
\(\chi_{1832}(525,\cdot)\)
\(\chi_{1832}(589,\cdot)\)
\(\chi_{1832}(597,\cdot)\)
\(\chi_{1832}(613,\cdot)\)
\(\chi_{1832}(621,\cdot)\)
\(\chi_{1832}(637,\cdot)\)
\(\chi_{1832}(677,\cdot)\)
\(\chi_{1832}(693,\cdot)\)
\(\chi_{1832}(725,\cdot)\)
\(\chi_{1832}(789,\cdot)\)
\(\chi_{1832}(797,\cdot)\)
\(\chi_{1832}(829,\cdot)\)
...
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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 };
\((1375,917,1609)\) → \((1,-1,e\left(\frac{155}{228}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1832 }(381, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{169}{228}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{49}{76}\right)\) |
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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)
