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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4699, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,23]))
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gp:[g,chi] = znchar(Mod(591, 4699))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4699.591");
| Modulus: | \(4699\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4699\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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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\(\chi_{4699}(110,\cdot)\)
\(\chi_{4699}(184,\cdot)\)
\(\chi_{4699}(332,\cdot)\)
\(\chi_{4699}(554,\cdot)\)
\(\chi_{4699}(591,\cdot)\)
\(\chi_{4699}(702,\cdot)\)
\(\chi_{4699}(776,\cdot)\)
\(\chi_{4699}(998,\cdot)\)
\(\chi_{4699}(1072,\cdot)\)
\(\chi_{4699}(1109,\cdot)\)
\(\chi_{4699}(1146,\cdot)\)
\(\chi_{4699}(1257,\cdot)\)
\(\chi_{4699}(1442,\cdot)\)
\(\chi_{4699}(1553,\cdot)\)
\(\chi_{4699}(1960,\cdot)\)
\(\chi_{4699}(1997,\cdot)\)
\(\chi_{4699}(2071,\cdot)\)
\(\chi_{4699}(2182,\cdot)\)
\(\chi_{4699}(2256,\cdot)\)
\(\chi_{4699}(2293,\cdot)\)
\(\chi_{4699}(2404,\cdot)\)
\(\chi_{4699}(2478,\cdot)\)
\(\chi_{4699}(2552,\cdot)\)
\(\chi_{4699}(2626,\cdot)\)
\(\chi_{4699}(2885,\cdot)\)
\(\chi_{4699}(3033,\cdot)\)
\(\chi_{4699}(3144,\cdot)\)
\(\chi_{4699}(3181,\cdot)\)
\(\chi_{4699}(3218,\cdot)\)
\(\chi_{4699}(3403,\cdot)\)
...
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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 };
\((890,2924)\) → \((-1,e\left(\frac{23}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4699 }(591, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{26}{63}\right)\) |
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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)
