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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3707, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([84,475]))
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gp:[g,chi] = znchar(Mod(112, 3707))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3707.112");
| Modulus: | \(3707\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3707\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(840\) |
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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);
|
\(\chi_{3707}(24,\cdot)\)
\(\chi_{3707}(28,\cdot)\)
\(\chi_{3707}(50,\cdot)\)
\(\chi_{3707}(63,\cdot)\)
\(\chi_{3707}(94,\cdot)\)
\(\chi_{3707}(95,\cdot)\)
\(\chi_{3707}(107,\cdot)\)
\(\chi_{3707}(112,\cdot)\)
\(\chi_{3707}(145,\cdot)\)
\(\chi_{3707}(149,\cdot)\)
\(\chi_{3707}(167,\cdot)\)
\(\chi_{3707}(182,\cdot)\)
\(\chi_{3707}(211,\cdot)\)
\(\chi_{3707}(222,\cdot)\)
\(\chi_{3707}(237,\cdot)\)
\(\chi_{3707}(259,\cdot)\)
\(\chi_{3707}(325,\cdot)\)
\(\chi_{3707}(349,\cdot)\)
\(\chi_{3707}(365,\cdot)\)
\(\chi_{3707}(387,\cdot)\)
\(\chi_{3707}(415,\cdot)\)
\(\chi_{3707}(431,\cdot)\)
\(\chi_{3707}(437,\cdot)\)
\(\chi_{3707}(486,\cdot)\)
\(\chi_{3707}(492,\cdot)\)
\(\chi_{3707}(519,\cdot)\)
\(\chi_{3707}(525,\cdot)\)
\(\chi_{3707}(567,\cdot)\)
\(\chi_{3707}(574,\cdot)\)
\(\chi_{3707}(579,\cdot)\)
...
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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 };
\((2697,1695)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{95}{168}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3707 }(112, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{17}{280}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) |
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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)
