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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9675, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([35,189,180]))
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gp:[g,chi] = znchar(Mod(3719, 9675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9675.3719");
| Modulus: | \(9675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{9675}(59,\cdot)\)
\(\chi_{9675}(164,\cdot)\)
\(\chi_{9675}(434,\cdot)\)
\(\chi_{9675}(704,\cdot)\)
\(\chi_{9675}(914,\cdot)\)
\(\chi_{9675}(1139,\cdot)\)
\(\chi_{9675}(1454,\cdot)\)
\(\chi_{9675}(1559,\cdot)\)
\(\chi_{9675}(1589,\cdot)\)
\(\chi_{9675}(1784,\cdot)\)
\(\chi_{9675}(1994,\cdot)\)
\(\chi_{9675}(2234,\cdot)\)
\(\chi_{9675}(2369,\cdot)\)
\(\chi_{9675}(2639,\cdot)\)
\(\chi_{9675}(3389,\cdot)\)
\(\chi_{9675}(3494,\cdot)\)
\(\chi_{9675}(3659,\cdot)\)
\(\chi_{9675}(3719,\cdot)\)
\(\chi_{9675}(3929,\cdot)\)
\(\chi_{9675}(4034,\cdot)\)
\(\chi_{9675}(4169,\cdot)\)
\(\chi_{9675}(4304,\cdot)\)
\(\chi_{9675}(4784,\cdot)\)
\(\chi_{9675}(5009,\cdot)\)
\(\chi_{9675}(5429,\cdot)\)
\(\chi_{9675}(5459,\cdot)\)
\(\chi_{9675}(5594,\cdot)\)
\(\chi_{9675}(5654,\cdot)\)
\(\chi_{9675}(5864,\cdot)\)
\(\chi_{9675}(5969,\cdot)\)
...
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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 };
\((7526,8902,5851)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 9675 }(3719, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) |
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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)
