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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16335, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([176,99,90]))
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gp:[g,chi] = znchar(Mod(34, 16335))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16335.34");
| Modulus: | \(16335\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(16335\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(198\) |
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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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\(\chi_{16335}(34,\cdot)\)
\(\chi_{16335}(529,\cdot)\)
\(\chi_{16335}(859,\cdot)\)
\(\chi_{16335}(1024,\cdot)\)
\(\chi_{16335}(1354,\cdot)\)
\(\chi_{16335}(1519,\cdot)\)
\(\chi_{16335}(1849,\cdot)\)
\(\chi_{16335}(2014,\cdot)\)
\(\chi_{16335}(2344,\cdot)\)
\(\chi_{16335}(2509,\cdot)\)
\(\chi_{16335}(2839,\cdot)\)
\(\chi_{16335}(3004,\cdot)\)
\(\chi_{16335}(3334,\cdot)\)
\(\chi_{16335}(3499,\cdot)\)
\(\chi_{16335}(3829,\cdot)\)
\(\chi_{16335}(4324,\cdot)\)
\(\chi_{16335}(4489,\cdot)\)
\(\chi_{16335}(4819,\cdot)\)
\(\chi_{16335}(4984,\cdot)\)
\(\chi_{16335}(5314,\cdot)\)
\(\chi_{16335}(5479,\cdot)\)
\(\chi_{16335}(5974,\cdot)\)
\(\chi_{16335}(6304,\cdot)\)
\(\chi_{16335}(6469,\cdot)\)
\(\chi_{16335}(6799,\cdot)\)
\(\chi_{16335}(6964,\cdot)\)
\(\chi_{16335}(7294,\cdot)\)
\(\chi_{16335}(7459,\cdot)\)
\(\chi_{16335}(7789,\cdot)\)
\(\chi_{16335}(7954,\cdot)\)
...
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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 };
\((3026,9802,3511)\) → \((e\left(\frac{8}{9}\right),-1,e\left(\frac{5}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 16335 }(34, a) \) |
\(1\) | \(1\) | \(e\left(\frac{167}{198}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{179}{198}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{103}{198}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{19}{198}\right)\) |
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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)
