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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8860, base_ring=CyclotomicField(884))
M = H._module
chi = DirichletCharacter(H, M([0,663,632]))
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gp:[g,chi] = znchar(Mod(333, 8860))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8860.333");
| Modulus: | \(8860\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2215\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(884\) |
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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: | no, induced from \(\chi_{2215}(333,\cdot)\) |
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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);
|
\(\chi_{8860}(17,\cdot)\)
\(\chi_{8860}(37,\cdot)\)
\(\chi_{8860}(77,\cdot)\)
\(\chi_{8860}(113,\cdot)\)
\(\chi_{8860}(117,\cdot)\)
\(\chi_{8860}(133,\cdot)\)
\(\chi_{8860}(137,\cdot)\)
\(\chi_{8860}(153,\cdot)\)
\(\chi_{8860}(177,\cdot)\)
\(\chi_{8860}(197,\cdot)\)
\(\chi_{8860}(213,\cdot)\)
\(\chi_{8860}(217,\cdot)\)
\(\chi_{8860}(233,\cdot)\)
\(\chi_{8860}(237,\cdot)\)
\(\chi_{8860}(253,\cdot)\)
\(\chi_{8860}(293,\cdot)\)
\(\chi_{8860}(333,\cdot)\)
\(\chi_{8860}(373,\cdot)\)
\(\chi_{8860}(393,\cdot)\)
\(\chi_{8860}(417,\cdot)\)
\(\chi_{8860}(437,\cdot)\)
\(\chi_{8860}(453,\cdot)\)
\(\chi_{8860}(457,\cdot)\)
\(\chi_{8860}(473,\cdot)\)
\(\chi_{8860}(533,\cdot)\)
\(\chi_{8860}(557,\cdot)\)
\(\chi_{8860}(573,\cdot)\)
\(\chi_{8860}(613,\cdot)\)
\(\chi_{8860}(617,\cdot)\)
\(\chi_{8860}(637,\cdot)\)
...
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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 };
\((4431,5317,5761)\) → \((1,-i,e\left(\frac{158}{221}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 8860 }(333, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{529}{884}\right)\) | \(e\left(\frac{671}{884}\right)\) | \(e\left(\frac{87}{442}\right)\) | \(e\left(\frac{200}{221}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{499}{884}\right)\) | \(e\left(\frac{211}{442}\right)\) | \(e\left(\frac{79}{221}\right)\) | \(e\left(\frac{297}{884}\right)\) | \(e\left(\frac{703}{884}\right)\) |
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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)
