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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9073, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([100,116]))
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gp:[g,chi] = znchar(Mod(2594, 9073))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9073.2594");
| Modulus: | \(9073\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9073\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(105\) |
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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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\(\chi_{9073}(52,\cdot)\)
\(\chi_{9073}(314,\cdot)\)
\(\chi_{9073}(841,\cdot)\)
\(\chi_{9073}(963,\cdot)\)
\(\chi_{9073}(1014,\cdot)\)
\(\chi_{9073}(1493,\cdot)\)
\(\chi_{9073}(1694,\cdot)\)
\(\chi_{9073}(1923,\cdot)\)
\(\chi_{9073}(1948,\cdot)\)
\(\chi_{9073}(1992,\cdot)\)
\(\chi_{9073}(2190,\cdot)\)
\(\chi_{9073}(2249,\cdot)\)
\(\chi_{9073}(2552,\cdot)\)
\(\chi_{9073}(2568,\cdot)\)
\(\chi_{9073}(2594,\cdot)\)
\(\chi_{9073}(2704,\cdot)\)
\(\chi_{9073}(2790,\cdot)\)
\(\chi_{9073}(2919,\cdot)\)
\(\chi_{9073}(2947,\cdot)\)
\(\chi_{9073}(3163,\cdot)\)
\(\chi_{9073}(3539,\cdot)\)
\(\chi_{9073}(3781,\cdot)\)
\(\chi_{9073}(3893,\cdot)\)
\(\chi_{9073}(4271,\cdot)\)
\(\chi_{9073}(4340,\cdot)\)
\(\chi_{9073}(4497,\cdot)\)
\(\chi_{9073}(4711,\cdot)\)
\(\chi_{9073}(5169,\cdot)\)
\(\chi_{9073}(5218,\cdot)\)
\(\chi_{9073}(5312,\cdot)\)
...
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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 };
\((4432,2323)\) → \((e\left(\frac{10}{21}\right),e\left(\frac{58}{105}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 9073 }(2594, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) |
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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)
