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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10085, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([189,26]))
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gp:[g,chi] = znchar(Mod(3313, 10085))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10085.3313");
| Modulus: | \(10085\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10085\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(252\) |
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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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\(\chi_{10085}(93,\cdot)\)
\(\chi_{10085}(682,\cdot)\)
\(\chi_{10085}(683,\cdot)\)
\(\chi_{10085}(698,\cdot)\)
\(\chi_{10085}(803,\cdot)\)
\(\chi_{10085}(887,\cdot)\)
\(\chi_{10085}(1828,\cdot)\)
\(\chi_{10085}(1878,\cdot)\)
\(\chi_{10085}(1967,\cdot)\)
\(\chi_{10085}(2138,\cdot)\)
\(\chi_{10085}(2183,\cdot)\)
\(\chi_{10085}(2232,\cdot)\)
\(\chi_{10085}(2413,\cdot)\)
\(\chi_{10085}(2527,\cdot)\)
\(\chi_{10085}(2562,\cdot)\)
\(\chi_{10085}(2602,\cdot)\)
\(\chi_{10085}(2628,\cdot)\)
\(\chi_{10085}(2648,\cdot)\)
\(\chi_{10085}(2682,\cdot)\)
\(\chi_{10085}(2842,\cdot)\)
\(\chi_{10085}(2927,\cdot)\)
\(\chi_{10085}(3138,\cdot)\)
\(\chi_{10085}(3222,\cdot)\)
\(\chi_{10085}(3302,\cdot)\)
\(\chi_{10085}(3312,\cdot)\)
\(\chi_{10085}(3313,\cdot)\)
\(\chi_{10085}(3453,\cdot)\)
\(\chi_{10085}(3472,\cdot)\)
\(\chi_{10085}(3508,\cdot)\)
\(\chi_{10085}(3512,\cdot)\)
...
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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 };
\((6052,6056)\) → \((-i,e\left(\frac{13}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 10085 }(3313, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{131}{252}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{185}{252}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{7}{36}\right)\) |
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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)
