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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3071, base_ring=CyclotomicField(246))
M = H._module
chi = DirichletCharacter(H, M([41,153]))
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gp:[g,chi] = znchar(Mod(138, 3071))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3071.138");
| Modulus: | \(3071\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3071\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(246\) |
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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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\(\chi_{3071}(85,\cdot)\)
\(\chi_{3071}(101,\cdot)\)
\(\chi_{3071}(122,\cdot)\)
\(\chi_{3071}(138,\cdot)\)
\(\chi_{3071}(159,\cdot)\)
\(\chi_{3071}(212,\cdot)\)
\(\chi_{3071}(233,\cdot)\)
\(\chi_{3071}(307,\cdot)\)
\(\chi_{3071}(323,\cdot)\)
\(\chi_{3071}(434,\cdot)\)
\(\chi_{3071}(471,\cdot)\)
\(\chi_{3071}(545,\cdot)\)
\(\chi_{3071}(603,\cdot)\)
\(\chi_{3071}(677,\cdot)\)
\(\chi_{3071}(714,\cdot)\)
\(\chi_{3071}(730,\cdot)\)
\(\chi_{3071}(767,\cdot)\)
\(\chi_{3071}(804,\cdot)\)
\(\chi_{3071}(862,\cdot)\)
\(\chi_{3071}(915,\cdot)\)
\(\chi_{3071}(952,\cdot)\)
\(\chi_{3071}(973,\cdot)\)
\(\chi_{3071}(989,\cdot)\)
\(\chi_{3071}(1010,\cdot)\)
\(\chi_{3071}(1063,\cdot)\)
\(\chi_{3071}(1084,\cdot)\)
\(\chi_{3071}(1121,\cdot)\)
\(\chi_{3071}(1137,\cdot)\)
\(\chi_{3071}(1158,\cdot)\)
\(\chi_{3071}(1269,\cdot)\)
...
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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 };
\((2740,334)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{51}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3071 }(138, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{97}{123}\right)\) | \(e\left(\frac{14}{123}\right)\) | \(e\left(\frac{71}{123}\right)\) | \(e\left(\frac{77}{123}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{38}{123}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{28}{123}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) |
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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)
