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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8424, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([54,54,44,9]))
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gp:[g,chi] = znchar(Mod(691, 8424))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8424.691");
| Modulus: | \(8424\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8424\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(108\) |
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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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\(\chi_{8424}(115,\cdot)\)
\(\chi_{8424}(643,\cdot)\)
\(\chi_{8424}(691,\cdot)\)
\(\chi_{8424}(787,\cdot)\)
\(\chi_{8424}(1051,\cdot)\)
\(\chi_{8424}(1579,\cdot)\)
\(\chi_{8424}(1627,\cdot)\)
\(\chi_{8424}(1723,\cdot)\)
\(\chi_{8424}(1987,\cdot)\)
\(\chi_{8424}(2515,\cdot)\)
\(\chi_{8424}(2563,\cdot)\)
\(\chi_{8424}(2659,\cdot)\)
\(\chi_{8424}(2923,\cdot)\)
\(\chi_{8424}(3451,\cdot)\)
\(\chi_{8424}(3499,\cdot)\)
\(\chi_{8424}(3595,\cdot)\)
\(\chi_{8424}(3859,\cdot)\)
\(\chi_{8424}(4387,\cdot)\)
\(\chi_{8424}(4435,\cdot)\)
\(\chi_{8424}(4531,\cdot)\)
\(\chi_{8424}(4795,\cdot)\)
\(\chi_{8424}(5323,\cdot)\)
\(\chi_{8424}(5371,\cdot)\)
\(\chi_{8424}(5467,\cdot)\)
\(\chi_{8424}(5731,\cdot)\)
\(\chi_{8424}(6259,\cdot)\)
\(\chi_{8424}(6307,\cdot)\)
\(\chi_{8424}(6403,\cdot)\)
\(\chi_{8424}(6667,\cdot)\)
\(\chi_{8424}(7195,\cdot)\)
...
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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 };
\((6319,4213,7697,3889)\) → \((-1,-1,e\left(\frac{11}{27}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8424 }(691, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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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)
