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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3925, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([117,275]))
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gp:[g,chi] = znchar(Mod(558, 3925))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3925.558");
| Modulus: | \(3925\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3925\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{3925}(53,\cdot)\)
\(\chi_{3925}(63,\cdot)\)
\(\chi_{3925}(83,\cdot)\)
\(\chi_{3925}(88,\cdot)\)
\(\chi_{3925}(123,\cdot)\)
\(\chi_{3925}(133,\cdot)\)
\(\chi_{3925}(137,\cdot)\)
\(\chi_{3925}(142,\cdot)\)
\(\chi_{3925}(162,\cdot)\)
\(\chi_{3925}(163,\cdot)\)
\(\chi_{3925}(183,\cdot)\)
\(\chi_{3925}(212,\cdot)\)
\(\chi_{3925}(217,\cdot)\)
\(\chi_{3925}(223,\cdot)\)
\(\chi_{3925}(227,\cdot)\)
\(\chi_{3925}(237,\cdot)\)
\(\chi_{3925}(242,\cdot)\)
\(\chi_{3925}(252,\cdot)\)
\(\chi_{3925}(253,\cdot)\)
\(\chi_{3925}(352,\cdot)\)
\(\chi_{3925}(387,\cdot)\)
\(\chi_{3925}(398,\cdot)\)
\(\chi_{3925}(428,\cdot)\)
\(\chi_{3925}(433,\cdot)\)
\(\chi_{3925}(453,\cdot)\)
\(\chi_{3925}(492,\cdot)\)
\(\chi_{3925}(533,\cdot)\)
\(\chi_{3925}(548,\cdot)\)
\(\chi_{3925}(558,\cdot)\)
\(\chi_{3925}(562,\cdot)\)
...
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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 };
\((2827,476)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{55}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3925 }(558, a) \) |
\(1\) | \(1\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{749}{780}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{641}{780}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{359}{390}\right)\) | \(e\left(\frac{53}{195}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) |
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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)
