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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5476, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([0,200]))
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gp:[g,chi] = znchar(Mod(1617, 5476))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5476.1617");
| Modulus: | \(5476\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1369\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(111\) |
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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 \(\chi_{1369}(248,\cdot)\) |
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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);
|
\(\chi_{5476}(121,\cdot)\)
\(\chi_{5476}(137,\cdot)\)
\(\chi_{5476}(269,\cdot)\)
\(\chi_{5476}(285,\cdot)\)
\(\chi_{5476}(417,\cdot)\)
\(\chi_{5476}(433,\cdot)\)
\(\chi_{5476}(565,\cdot)\)
\(\chi_{5476}(713,\cdot)\)
\(\chi_{5476}(729,\cdot)\)
\(\chi_{5476}(861,\cdot)\)
\(\chi_{5476}(877,\cdot)\)
\(\chi_{5476}(1009,\cdot)\)
\(\chi_{5476}(1025,\cdot)\)
\(\chi_{5476}(1157,\cdot)\)
\(\chi_{5476}(1173,\cdot)\)
\(\chi_{5476}(1305,\cdot)\)
\(\chi_{5476}(1321,\cdot)\)
\(\chi_{5476}(1453,\cdot)\)
\(\chi_{5476}(1469,\cdot)\)
\(\chi_{5476}(1601,\cdot)\)
\(\chi_{5476}(1617,\cdot)\)
\(\chi_{5476}(1749,\cdot)\)
\(\chi_{5476}(1765,\cdot)\)
\(\chi_{5476}(1897,\cdot)\)
\(\chi_{5476}(1913,\cdot)\)
\(\chi_{5476}(2045,\cdot)\)
\(\chi_{5476}(2061,\cdot)\)
\(\chi_{5476}(2193,\cdot)\)
\(\chi_{5476}(2209,\cdot)\)
\(\chi_{5476}(2341,\cdot)\)
...
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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 };
\((2739,4109)\) → \((1,e\left(\frac{100}{111}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 5476 }(1617, a) \) |
\(1\) | \(1\) | \(e\left(\frac{98}{111}\right)\) | \(e\left(\frac{110}{111}\right)\) | \(e\left(\frac{77}{111}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{26}{111}\right)\) | \(e\left(\frac{97}{111}\right)\) | \(e\left(\frac{19}{111}\right)\) | \(e\left(\frac{59}{111}\right)\) | \(e\left(\frac{64}{111}\right)\) |
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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)
