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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26195, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([195,265,234]))
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gp:[g,chi] = znchar(Mod(1982, 26195))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26195.1982");
| Modulus: | \(26195\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(26195\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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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\(\chi_{26195}(457,\cdot)\)
\(\chi_{26195}(618,\cdot)\)
\(\chi_{26195}(1007,\cdot)\)
\(\chi_{26195}(1038,\cdot)\)
\(\chi_{26195}(1267,\cdot)\)
\(\chi_{26195}(1298,\cdot)\)
\(\chi_{26195}(1627,\cdot)\)
\(\chi_{26195}(1658,\cdot)\)
\(\chi_{26195}(1852,\cdot)\)
\(\chi_{26195}(1883,\cdot)\)
\(\chi_{26195}(1887,\cdot)\)
\(\chi_{26195}(1918,\cdot)\)
\(\chi_{26195}(1982,\cdot)\)
\(\chi_{26195}(2013,\cdot)\)
\(\chi_{26195}(2472,\cdot)\)
\(\chi_{26195}(2503,\cdot)\)
\(\chi_{26195}(2602,\cdot)\)
\(\chi_{26195}(2633,\cdot)\)
\(\chi_{26195}(3022,\cdot)\)
\(\chi_{26195}(3053,\cdot)\)
\(\chi_{26195}(3282,\cdot)\)
\(\chi_{26195}(3313,\cdot)\)
\(\chi_{26195}(3642,\cdot)\)
\(\chi_{26195}(3673,\cdot)\)
\(\chi_{26195}(3867,\cdot)\)
\(\chi_{26195}(3898,\cdot)\)
\(\chi_{26195}(3902,\cdot)\)
\(\chi_{26195}(3933,\cdot)\)
\(\chi_{26195}(3997,\cdot)\)
\(\chi_{26195}(4028,\cdot)\)
...
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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 };
\((20957,1861,6761)\) → \((i,e\left(\frac{53}{156}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 26195 }(1982, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{139}{780}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{1}{390}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{139}{390}\right)\) | \(e\left(\frac{697}{780}\right)\) | \(e\left(\frac{197}{260}\right)\) | \(e\left(\frac{103}{130}\right)\) |
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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)
