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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41382, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,28,55]))
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gp:[g,chi] = znchar(Mod(14743, 41382))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41382.14743");
| Modulus: | \(41382\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2299\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{2299}(949,\cdot)\) |
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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);
|
\(\chi_{41382}(37,\cdot)\)
\(\chi_{41382}(379,\cdot)\)
\(\chi_{41382}(1747,\cdot)\)
\(\chi_{41382}(3457,\cdot)\)
\(\chi_{41382}(3799,\cdot)\)
\(\chi_{41382}(5509,\cdot)\)
\(\chi_{41382}(7219,\cdot)\)
\(\chi_{41382}(7561,\cdot)\)
\(\chi_{41382}(7903,\cdot)\)
\(\chi_{41382}(9271,\cdot)\)
\(\chi_{41382}(10981,\cdot)\)
\(\chi_{41382}(11323,\cdot)\)
\(\chi_{41382}(11665,\cdot)\)
\(\chi_{41382}(13033,\cdot)\)
\(\chi_{41382}(14743,\cdot)\)
\(\chi_{41382}(15427,\cdot)\)
\(\chi_{41382}(16795,\cdot)\)
\(\chi_{41382}(18505,\cdot)\)
\(\chi_{41382}(18847,\cdot)\)
\(\chi_{41382}(19189,\cdot)\)
\(\chi_{41382}(20557,\cdot)\)
\(\chi_{41382}(22609,\cdot)\)
\(\chi_{41382}(22951,\cdot)\)
\(\chi_{41382}(24319,\cdot)\)
\(\chi_{41382}(26029,\cdot)\)
\(\chi_{41382}(26371,\cdot)\)
\(\chi_{41382}(26713,\cdot)\)
\(\chi_{41382}(29791,\cdot)\)
\(\chi_{41382}(30133,\cdot)\)
\(\chi_{41382}(30475,\cdot)\)
...
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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 };
\((36785,7867,17425)\) → \((1,e\left(\frac{14}{55}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 41382 }(14743, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) |
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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)
