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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40051, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([123,82]))
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gp:[g,chi] = znchar(Mod(11937, 40051))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40051.11937");
| Modulus: | \(40051\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(40051\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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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\(\chi_{40051}(756,\cdot)\)
\(\chi_{40051}(833,\cdot)\)
\(\chi_{40051}(953,\cdot)\)
\(\chi_{40051}(1333,\cdot)\)
\(\chi_{40051}(1557,\cdot)\)
\(\chi_{40051}(1850,\cdot)\)
\(\chi_{40051}(3251,\cdot)\)
\(\chi_{40051}(4385,\cdot)\)
\(\chi_{40051}(4715,\cdot)\)
\(\chi_{40051}(5914,\cdot)\)
\(\chi_{40051}(6342,\cdot)\)
\(\chi_{40051}(7064,\cdot)\)
\(\chi_{40051}(7279,\cdot)\)
\(\chi_{40051}(8509,\cdot)\)
\(\chi_{40051}(8571,\cdot)\)
\(\chi_{40051}(8632,\cdot)\)
\(\chi_{40051}(8982,\cdot)\)
\(\chi_{40051}(9213,\cdot)\)
\(\chi_{40051}(9533,\cdot)\)
\(\chi_{40051}(11415,\cdot)\)
\(\chi_{40051}(11574,\cdot)\)
\(\chi_{40051}(11937,\cdot)\)
\(\chi_{40051}(13246,\cdot)\)
\(\chi_{40051}(13361,\cdot)\)
\(\chi_{40051}(13362,\cdot)\)
\(\chi_{40051}(13978,\cdot)\)
\(\chi_{40051}(14011,\cdot)\)
\(\chi_{40051}(14758,\cdot)\)
\(\chi_{40051}(15078,\cdot)\)
\(\chi_{40051}(15644,\cdot)\)
...
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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 };
\((11255,17546)\) → \((e\left(\frac{41}{110}\right),e\left(\frac{41}{165}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 40051 }(11937, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) |
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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)
