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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3336, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
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gp:[g,chi] = znchar(Mod(697, 3336))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3336.697");
| Modulus: | \(3336\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(139\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{139}(2,\cdot)\) |
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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: | 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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\(\chi_{3336}(73,\cdot)\)
\(\chi_{3336}(241,\cdot)\)
\(\chi_{3336}(265,\cdot)\)
\(\chi_{3336}(457,\cdot)\)
\(\chi_{3336}(505,\cdot)\)
\(\chi_{3336}(577,\cdot)\)
\(\chi_{3336}(649,\cdot)\)
\(\chi_{3336}(697,\cdot)\)
\(\chi_{3336}(721,\cdot)\)
\(\chi_{3336}(745,\cdot)\)
\(\chi_{3336}(793,\cdot)\)
\(\chi_{3336}(985,\cdot)\)
\(\chi_{3336}(1081,\cdot)\)
\(\chi_{3336}(1105,\cdot)\)
\(\chi_{3336}(1129,\cdot)\)
\(\chi_{3336}(1273,\cdot)\)
\(\chi_{3336}(1321,\cdot)\)
\(\chi_{3336}(1393,\cdot)\)
\(\chi_{3336}(1513,\cdot)\)
\(\chi_{3336}(1561,\cdot)\)
\(\chi_{3336}(1585,\cdot)\)
\(\chi_{3336}(1633,\cdot)\)
\(\chi_{3336}(1657,\cdot)\)
\(\chi_{3336}(1729,\cdot)\)
\(\chi_{3336}(1753,\cdot)\)
\(\chi_{3336}(1777,\cdot)\)
\(\chi_{3336}(1825,\cdot)\)
\(\chi_{3336}(1897,\cdot)\)
\(\chi_{3336}(1921,\cdot)\)
\(\chi_{3336}(2065,\cdot)\)
...
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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 };
\((2503,1669,2225,697)\) → \((1,1,1,e\left(\frac{1}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3336 }(697, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{28}{69}\right)\) |
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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)
