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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61969, base_ring=CyclotomicField(9990))
M = H._module
chi = DirichletCharacter(H, M([4329,6010]))
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gp:[g,chi] = znchar(Mod(16454, 61969))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61969.16454");
| Modulus: | \(61969\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(61969\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(9990\) |
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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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\(\chi_{61969}(11,\cdot)\)
\(\chi_{61969}(21,\cdot)\)
\(\chi_{61969}(22,\cdot)\)
\(\chi_{61969}(42,\cdot)\)
\(\chi_{61969}(44,\cdot)\)
\(\chi_{61969}(53,\cdot)\)
\(\chi_{61969}(55,\cdot)\)
\(\chi_{61969}(79,\cdot)\)
\(\chi_{61969}(84,\cdot)\)
\(\chi_{61969}(105,\cdot)\)
\(\chi_{61969}(106,\cdot)\)
\(\chi_{61969}(110,\cdot)\)
\(\chi_{61969}(117,\cdot)\)
\(\chi_{61969}(137,\cdot)\)
\(\chi_{61969}(141,\cdot)\)
\(\chi_{61969}(158,\cdot)\)
\(\chi_{61969}(168,\cdot)\)
\(\chi_{61969}(210,\cdot)\)
\(\chi_{61969}(220,\cdot)\)
\(\chi_{61969}(234,\cdot)\)
\(\chi_{61969}(251,\cdot)\)
\(\chi_{61969}(265,\cdot)\)
\(\chi_{61969}(282,\cdot)\)
\(\chi_{61969}(327,\cdot)\)
\(\chi_{61969}(352,\cdot)\)
\(\chi_{61969}(389,\cdot)\)
\(\chi_{61969}(393,\cdot)\)
\(\chi_{61969}(420,\cdot)\)
\(\chi_{61969}(424,\cdot)\)
\(\chi_{61969}(451,\cdot)\)
...
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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 };
\((53974,7999)\) → \((e\left(\frac{13}{30}\right),e\left(\frac{601}{999}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 61969 }(16454, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{361}{1665}\right)\) | \(e\left(\frac{349}{9990}\right)\) | \(e\left(\frac{722}{1665}\right)\) | \(e\left(\frac{910}{999}\right)\) | \(e\left(\frac{503}{1998}\right)\) | \(e\left(\frac{1042}{1665}\right)\) | \(e\left(\frac{361}{555}\right)\) | \(e\left(\frac{349}{4995}\right)\) | \(e\left(\frac{638}{4995}\right)\) | \(e\left(\frac{1807}{9990}\right)\) |
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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)
