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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85547, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([110,168,121]))
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gp:[g,chi] = znchar(Mod(5725, 85547))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("85547.5725");
| Modulus: | \(85547\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(85547\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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);
|
\(\chi_{85547}(335,\cdot)\)
\(\chi_{85547}(1357,\cdot)\)
\(\chi_{85547}(2484,\cdot)\)
\(\chi_{85547}(2869,\cdot)\)
\(\chi_{85547}(4304,\cdot)\)
\(\chi_{85547}(5725,\cdot)\)
\(\chi_{85547}(5998,\cdot)\)
\(\chi_{85547}(6824,\cdot)\)
\(\chi_{85547}(8112,\cdot)\)
\(\chi_{85547}(9134,\cdot)\)
\(\chi_{85547}(10261,\cdot)\)
\(\chi_{85547}(10646,\cdot)\)
\(\chi_{85547}(12081,\cdot)\)
\(\chi_{85547}(13502,\cdot)\)
\(\chi_{85547}(13775,\cdot)\)
\(\chi_{85547}(15889,\cdot)\)
\(\chi_{85547}(16911,\cdot)\)
\(\chi_{85547}(18423,\cdot)\)
\(\chi_{85547}(19858,\cdot)\)
\(\chi_{85547}(21279,\cdot)\)
\(\chi_{85547}(21552,\cdot)\)
\(\chi_{85547}(22378,\cdot)\)
\(\chi_{85547}(23666,\cdot)\)
\(\chi_{85547}(24688,\cdot)\)
\(\chi_{85547}(25815,\cdot)\)
\(\chi_{85547}(26200,\cdot)\)
\(\chi_{85547}(27635,\cdot)\)
\(\chi_{85547}(29056,\cdot)\)
\(\chi_{85547}(29329,\cdot)\)
\(\chi_{85547}(30155,\cdot)\)
...
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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 };
\((61106,49491,48280)\) → \((-1,e\left(\frac{42}{55}\right),e\left(\frac{11}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 85547 }(5725, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{51}{55}\right)\) |
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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)
