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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20576, base_ring=CyclotomicField(428))
M = H._module
chi = DirichletCharacter(H, M([214,321,204]))
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gp:[g,chi] = znchar(Mod(343, 20576))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20576.343");
| Modulus: | \(20576\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10288\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(428\) |
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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 \(\chi_{10288}(2915,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{20576}(135,\cdot)\)
\(\chi_{20576}(167,\cdot)\)
\(\chi_{20576}(215,\cdot)\)
\(\chi_{20576}(231,\cdot)\)
\(\chi_{20576}(343,\cdot)\)
\(\chi_{20576}(359,\cdot)\)
\(\chi_{20576}(375,\cdot)\)
\(\chi_{20576}(471,\cdot)\)
\(\chi_{20576}(535,\cdot)\)
\(\chi_{20576}(631,\cdot)\)
\(\chi_{20576}(647,\cdot)\)
\(\chi_{20576}(679,\cdot)\)
\(\chi_{20576}(743,\cdot)\)
\(\chi_{20576}(967,\cdot)\)
\(\chi_{20576}(1159,\cdot)\)
\(\chi_{20576}(1367,\cdot)\)
\(\chi_{20576}(1415,\cdot)\)
\(\chi_{20576}(1447,\cdot)\)
\(\chi_{20576}(1479,\cdot)\)
\(\chi_{20576}(1495,\cdot)\)
\(\chi_{20576}(1511,\cdot)\)
\(\chi_{20576}(1527,\cdot)\)
\(\chi_{20576}(1543,\cdot)\)
\(\chi_{20576}(1607,\cdot)\)
\(\chi_{20576}(1671,\cdot)\)
\(\chi_{20576}(1767,\cdot)\)
\(\chi_{20576}(1879,\cdot)\)
\(\chi_{20576}(1911,\cdot)\)
\(\chi_{20576}(1927,\cdot)\)
\(\chi_{20576}(2071,\cdot)\)
...
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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 };
\((6431,7717,11585)\) → \((-1,-i,e\left(\frac{51}{107}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 20576 }(343, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{428}\right)\) | \(e\left(\frac{265}{428}\right)\) | \(e\left(\frac{66}{107}\right)\) | \(e\left(\frac{13}{214}\right)\) | \(e\left(\frac{311}{428}\right)\) | \(e\left(\frac{91}{428}\right)\) | \(e\left(\frac{139}{214}\right)\) | \(e\left(\frac{49}{107}\right)\) | \(e\left(\frac{281}{428}\right)\) | \(e\left(\frac{277}{428}\right)\) |
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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)
