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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([9657,4295]))
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gp:[g,chi] = znchar(Mod(796, 61969))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61969.796");
| 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: | 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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\(\chi_{61969}(17,\cdot)\)
\(\chi_{61969}(34,\cdot)\)
\(\chi_{61969}(127,\cdot)\)
\(\chi_{61969}(135,\cdot)\)
\(\chi_{61969}(136,\cdot)\)
\(\chi_{61969}(199,\cdot)\)
\(\chi_{61969}(269,\cdot)\)
\(\chi_{61969}(270,\cdot)\)
\(\chi_{61969}(272,\cdot)\)
\(\chi_{61969}(425,\cdot)\)
\(\chi_{61969}(437,\cdot)\)
\(\chi_{61969}(445,\cdot)\)
\(\chi_{61969}(507,\cdot)\)
\(\chi_{61969}(508,\cdot)\)
\(\chi_{61969}(538,\cdot)\)
\(\chi_{61969}(540,\cdot)\)
\(\chi_{61969}(544,\cdot)\)
\(\chi_{61969}(611,\cdot)\)
\(\chi_{61969}(675,\cdot)\)
\(\chi_{61969}(699,\cdot)\)
\(\chi_{61969}(703,\cdot)\)
\(\chi_{61969}(796,\cdot)\)
\(\chi_{61969}(850,\cdot)\)
\(\chi_{61969}(890,\cdot)\)
\(\chi_{61969}(910,\cdot)\)
\(\chi_{61969}(921,\cdot)\)
\(\chi_{61969}(942,\cdot)\)
\(\chi_{61969}(947,\cdot)\)
\(\chi_{61969}(995,\cdot)\)
\(\chi_{61969}(1014,\cdot)\)
...
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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 };
\((53974,7999)\) → \((e\left(\frac{29}{30}\right),e\left(\frac{859}{1998}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 61969 }(796, a) \) |
\(1\) | \(1\) | \(e\left(\frac{773}{1665}\right)\) | \(e\left(\frac{1981}{4995}\right)\) | \(e\left(\frac{1546}{1665}\right)\) | \(e\left(\frac{647}{999}\right)\) | \(e\left(\frac{860}{999}\right)\) | \(e\left(\frac{67}{3330}\right)\) | \(e\left(\frac{218}{555}\right)\) | \(e\left(\frac{3962}{4995}\right)\) | \(e\left(\frac{559}{4995}\right)\) | \(e\left(\frac{3611}{9990}\right)\) |
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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)
