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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2130, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([0,35,76]))
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gp:[g,chi] = znchar(Mod(367, 2130))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2130.367");
| Modulus: | \(2130\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(355\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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 \(\chi_{355}(12,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{2130}(43,\cdot)\)
\(\chi_{2130}(73,\cdot)\)
\(\chi_{2130}(157,\cdot)\)
\(\chi_{2130}(217,\cdot)\)
\(\chi_{2130}(223,\cdot)\)
\(\chi_{2130}(253,\cdot)\)
\(\chi_{2130}(277,\cdot)\)
\(\chi_{2130}(313,\cdot)\)
\(\chi_{2130}(367,\cdot)\)
\(\chi_{2130}(373,\cdot)\)
\(\chi_{2130}(547,\cdot)\)
\(\chi_{2130}(577,\cdot)\)
\(\chi_{2130}(583,\cdot)\)
\(\chi_{2130}(643,\cdot)\)
\(\chi_{2130}(697,\cdot)\)
\(\chi_{2130}(703,\cdot)\)
\(\chi_{2130}(787,\cdot)\)
\(\chi_{2130}(793,\cdot)\)
\(\chi_{2130}(817,\cdot)\)
\(\chi_{2130}(973,\cdot)\)
\(\chi_{2130}(997,\cdot)\)
\(\chi_{2130}(1003,\cdot)\)
\(\chi_{2130}(1123,\cdot)\)
\(\chi_{2130}(1213,\cdot)\)
\(\chi_{2130}(1243,\cdot)\)
\(\chi_{2130}(1267,\cdot)\)
\(\chi_{2130}(1297,\cdot)\)
\(\chi_{2130}(1327,\cdot)\)
\(\chi_{2130}(1357,\cdot)\)
\(\chi_{2130}(1387,\cdot)\)
...
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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 };
\((1421,427,1711)\) → \((1,i,e\left(\frac{19}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2130 }(367, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) |
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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)
