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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4500, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,250,273]))
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gp:[g,chi] = znchar(Mod(1823, 4500))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4500.1823");
| Modulus: | \(4500\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4500\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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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\(\chi_{4500}(23,\cdot)\)
\(\chi_{4500}(47,\cdot)\)
\(\chi_{4500}(83,\cdot)\)
\(\chi_{4500}(167,\cdot)\)
\(\chi_{4500}(203,\cdot)\)
\(\chi_{4500}(227,\cdot)\)
\(\chi_{4500}(263,\cdot)\)
\(\chi_{4500}(347,\cdot)\)
\(\chi_{4500}(383,\cdot)\)
\(\chi_{4500}(527,\cdot)\)
\(\chi_{4500}(563,\cdot)\)
\(\chi_{4500}(587,\cdot)\)
\(\chi_{4500}(623,\cdot)\)
\(\chi_{4500}(767,\cdot)\)
\(\chi_{4500}(803,\cdot)\)
\(\chi_{4500}(887,\cdot)\)
\(\chi_{4500}(923,\cdot)\)
\(\chi_{4500}(947,\cdot)\)
\(\chi_{4500}(983,\cdot)\)
\(\chi_{4500}(1067,\cdot)\)
\(\chi_{4500}(1103,\cdot)\)
\(\chi_{4500}(1127,\cdot)\)
\(\chi_{4500}(1163,\cdot)\)
\(\chi_{4500}(1247,\cdot)\)
\(\chi_{4500}(1283,\cdot)\)
\(\chi_{4500}(1427,\cdot)\)
\(\chi_{4500}(1463,\cdot)\)
\(\chi_{4500}(1487,\cdot)\)
\(\chi_{4500}(1523,\cdot)\)
\(\chi_{4500}(1667,\cdot)\)
...
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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 };
\((2251,1001,2377)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{91}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4500 }(1823, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{47}{300}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{263}{300}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{31}{150}\right)\) |
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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)
