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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64675, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([77,55,80]))
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gp:[g,chi] = znchar(Mod(35684, 64675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64675.35684");
| Modulus: | \(64675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(64675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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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\(\chi_{64675}(519,\cdot)\)
\(\chi_{64675}(3444,\cdot)\)
\(\chi_{64675}(4094,\cdot)\)
\(\chi_{64675}(6109,\cdot)\)
\(\chi_{64675}(6629,\cdot)\)
\(\chi_{64675}(6954,\cdot)\)
\(\chi_{64675}(7864,\cdot)\)
\(\chi_{64675}(9814,\cdot)\)
\(\chi_{64675}(13259,\cdot)\)
\(\chi_{64675}(13454,\cdot)\)
\(\chi_{64675}(16379,\cdot)\)
\(\chi_{64675}(17029,\cdot)\)
\(\chi_{64675}(19044,\cdot)\)
\(\chi_{64675}(19564,\cdot)\)
\(\chi_{64675}(19889,\cdot)\)
\(\chi_{64675}(21709,\cdot)\)
\(\chi_{64675}(26194,\cdot)\)
\(\chi_{64675}(26389,\cdot)\)
\(\chi_{64675}(29314,\cdot)\)
\(\chi_{64675}(29964,\cdot)\)
\(\chi_{64675}(31979,\cdot)\)
\(\chi_{64675}(33734,\cdot)\)
\(\chi_{64675}(34644,\cdot)\)
\(\chi_{64675}(35684,\cdot)\)
\(\chi_{64675}(39129,\cdot)\)
\(\chi_{64675}(44914,\cdot)\)
\(\chi_{64675}(45434,\cdot)\)
\(\chi_{64675}(45759,\cdot)\)
\(\chi_{64675}(46669,\cdot)\)
\(\chi_{64675}(47579,\cdot)\)
...
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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 };
\((59502,14926,45176)\) → \((e\left(\frac{7}{10}\right),-1,e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 64675 }(35684, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) |
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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)
