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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45738, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([44,0,18]))
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gp:[g,chi] = znchar(Mod(6469, 45738))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45738.6469");
| Modulus: | \(45738\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3267\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(99\) |
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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 \(\chi_{3267}(3202,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{45738}(463,\cdot)\)
\(\chi_{45738}(925,\cdot)\)
\(\chi_{45738}(1849,\cdot)\)
\(\chi_{45738}(2311,\cdot)\)
\(\chi_{45738}(3235,\cdot)\)
\(\chi_{45738}(3697,\cdot)\)
\(\chi_{45738}(4621,\cdot)\)
\(\chi_{45738}(6007,\cdot)\)
\(\chi_{45738}(6469,\cdot)\)
\(\chi_{45738}(7393,\cdot)\)
\(\chi_{45738}(7855,\cdot)\)
\(\chi_{45738}(8779,\cdot)\)
\(\chi_{45738}(9241,\cdot)\)
\(\chi_{45738}(10627,\cdot)\)
\(\chi_{45738}(11551,\cdot)\)
\(\chi_{45738}(12013,\cdot)\)
\(\chi_{45738}(12937,\cdot)\)
\(\chi_{45738}(13399,\cdot)\)
\(\chi_{45738}(14323,\cdot)\)
\(\chi_{45738}(14785,\cdot)\)
\(\chi_{45738}(15709,\cdot)\)
\(\chi_{45738}(16171,\cdot)\)
\(\chi_{45738}(17095,\cdot)\)
\(\chi_{45738}(17557,\cdot)\)
\(\chi_{45738}(18481,\cdot)\)
\(\chi_{45738}(18943,\cdot)\)
\(\chi_{45738}(19867,\cdot)\)
\(\chi_{45738}(21253,\cdot)\)
\(\chi_{45738}(21715,\cdot)\)
\(\chi_{45738}(22639,\cdot)\)
...
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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 };
\((38963,19603,42715)\) → \((e\left(\frac{2}{9}\right),1,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 45738 }(6469, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{86}{99}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
