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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7525, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([42,70,100]))
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gp:[g,chi] = znchar(Mod(4916, 7525))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7525.4916");
| Modulus: | \(7525\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7525\) |
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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);
|
\(\chi_{7525}(81,\cdot)\)
\(\chi_{7525}(541,\cdot)\)
\(\chi_{7525}(611,\cdot)\)
\(\chi_{7525}(711,\cdot)\)
\(\chi_{7525}(956,\cdot)\)
\(\chi_{7525}(961,\cdot)\)
\(\chi_{7525}(1271,\cdot)\)
\(\chi_{7525}(1346,\cdot)\)
\(\chi_{7525}(1586,\cdot)\)
\(\chi_{7525}(1906,\cdot)\)
\(\chi_{7525}(2046,\cdot)\)
\(\chi_{7525}(2081,\cdot)\)
\(\chi_{7525}(2116,\cdot)\)
\(\chi_{7525}(2181,\cdot)\)
\(\chi_{7525}(2216,\cdot)\)
\(\chi_{7525}(2461,\cdot)\)
\(\chi_{7525}(2466,\cdot)\)
\(\chi_{7525}(2706,\cdot)\)
\(\chi_{7525}(3091,\cdot)\)
\(\chi_{7525}(3411,\cdot)\)
\(\chi_{7525}(3586,\cdot)\)
\(\chi_{7525}(3621,\cdot)\)
\(\chi_{7525}(3686,\cdot)\)
\(\chi_{7525}(3721,\cdot)\)
\(\chi_{7525}(3966,\cdot)\)
\(\chi_{7525}(3971,\cdot)\)
\(\chi_{7525}(4211,\cdot)\)
\(\chi_{7525}(4281,\cdot)\)
\(\chi_{7525}(4356,\cdot)\)
\(\chi_{7525}(4596,\cdot)\)
...
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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 };
\((302,4301,2626)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right),e\left(\frac{10}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 7525 }(4916, a) \) |
\(1\) | \(1\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) |
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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)
