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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2955, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([98,49,5]))
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gp:[g,chi] = znchar(Mod(32, 2955))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2955.32");
| Modulus: | \(2955\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2955\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(196\) |
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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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\(\chi_{2955}(2,\cdot)\)
\(\chi_{2955}(8,\cdot)\)
\(\chi_{2955}(32,\cdot)\)
\(\chi_{2955}(38,\cdot)\)
\(\chi_{2955}(122,\cdot)\)
\(\chi_{2955}(152,\cdot)\)
\(\chi_{2955}(167,\cdot)\)
\(\chi_{2955}(218,\cdot)\)
\(\chi_{2955}(263,\cdot)\)
\(\chi_{2955}(338,\cdot)\)
\(\chi_{2955}(377,\cdot)\)
\(\chi_{2955}(383,\cdot)\)
\(\chi_{2955}(407,\cdot)\)
\(\chi_{2955}(452,\cdot)\)
\(\chi_{2955}(467,\cdot)\)
\(\chi_{2955}(473,\cdot)\)
\(\chi_{2955}(488,\cdot)\)
\(\chi_{2955}(497,\cdot)\)
\(\chi_{2955}(512,\cdot)\)
\(\chi_{2955}(518,\cdot)\)
\(\chi_{2955}(533,\cdot)\)
\(\chi_{2955}(578,\cdot)\)
\(\chi_{2955}(602,\cdot)\)
\(\chi_{2955}(608,\cdot)\)
\(\chi_{2955}(647,\cdot)\)
\(\chi_{2955}(722,\cdot)\)
\(\chi_{2955}(767,\cdot)\)
\(\chi_{2955}(818,\cdot)\)
\(\chi_{2955}(833,\cdot)\)
\(\chi_{2955}(863,\cdot)\)
...
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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 };
\((986,592,1381)\) → \((-1,i,e\left(\frac{5}{196}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2955 }(32, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{47}{196}\right)\) | \(e\left(\frac{19}{49}\right)\) | \(-i\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
