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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7839, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([110,99,96]))
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gp:[g,chi] = znchar(Mod(4685, 7839))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7839.4685");
| Modulus: | \(7839\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7839\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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);
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\(\chi_{7839}(551,\cdot)\)
\(\chi_{7839}(866,\cdot)\)
\(\chi_{7839}(1019,\cdot)\)
\(\chi_{7839}(1136,\cdot)\)
\(\chi_{7839}(1958,\cdot)\)
\(\chi_{7839}(2072,\cdot)\)
\(\chi_{7839}(2153,\cdot)\)
\(\chi_{7839}(2270,\cdot)\)
\(\chi_{7839}(2426,\cdot)\)
\(\chi_{7839}(2504,\cdot)\)
\(\chi_{7839}(2543,\cdot)\)
\(\chi_{7839}(2972,\cdot)\)
\(\chi_{7839}(3164,\cdot)\)
\(\chi_{7839}(3323,\cdot)\)
\(\chi_{7839}(3359,\cdot)\)
\(\chi_{7839}(3476,\cdot)\)
\(\chi_{7839}(3479,\cdot)\)
\(\chi_{7839}(3632,\cdot)\)
\(\chi_{7839}(3710,\cdot)\)
\(\chi_{7839}(3749,\cdot)\)
\(\chi_{7839}(3908,\cdot)\)
\(\chi_{7839}(4178,\cdot)\)
\(\chi_{7839}(4529,\cdot)\)
\(\chi_{7839}(4685,\cdot)\)
\(\chi_{7839}(4766,\cdot)\)
\(\chi_{7839}(4883,\cdot)\)
\(\chi_{7839}(5114,\cdot)\)
\(\chi_{7839}(5117,\cdot)\)
\(\chi_{7839}(5585,\cdot)\)
\(\chi_{7839}(5936,\cdot)\)
...
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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 };
\((3485,4825,7372)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 7839 }(4685, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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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)
