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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2704, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,117,22]))
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gp:[g,chi] = znchar(Mod(1245, 2704))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2704.1245");
| Modulus: | \(2704\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2704\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{2704}(69,\cdot)\)
\(\chi_{2704}(101,\cdot)\)
\(\chi_{2704}(173,\cdot)\)
\(\chi_{2704}(205,\cdot)\)
\(\chi_{2704}(277,\cdot)\)
\(\chi_{2704}(309,\cdot)\)
\(\chi_{2704}(381,\cdot)\)
\(\chi_{2704}(413,\cdot)\)
\(\chi_{2704}(517,\cdot)\)
\(\chi_{2704}(589,\cdot)\)
\(\chi_{2704}(621,\cdot)\)
\(\chi_{2704}(693,\cdot)\)
\(\chi_{2704}(725,\cdot)\)
\(\chi_{2704}(797,\cdot)\)
\(\chi_{2704}(829,\cdot)\)
\(\chi_{2704}(901,\cdot)\)
\(\chi_{2704}(933,\cdot)\)
\(\chi_{2704}(1005,\cdot)\)
\(\chi_{2704}(1109,\cdot)\)
\(\chi_{2704}(1141,\cdot)\)
\(\chi_{2704}(1213,\cdot)\)
\(\chi_{2704}(1245,\cdot)\)
\(\chi_{2704}(1317,\cdot)\)
\(\chi_{2704}(1349,\cdot)\)
\(\chi_{2704}(1421,\cdot)\)
\(\chi_{2704}(1453,\cdot)\)
\(\chi_{2704}(1525,\cdot)\)
\(\chi_{2704}(1557,\cdot)\)
\(\chi_{2704}(1629,\cdot)\)
\(\chi_{2704}(1661,\cdot)\)
...
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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 };
\((2367,677,1185)\) → \((1,-i,e\left(\frac{11}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2704 }(1245, a) \) |
\(1\) | \(1\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
