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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35525, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([63,20,135]))
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gp:[g,chi] = znchar(Mod(10714, 35525))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35525.10714");
| Modulus: | \(35525\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(35525\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{35525}(109,\cdot)\)
\(\chi_{35525}(2034,\cdot)\)
\(\chi_{35525}(3194,\cdot)\)
\(\chi_{35525}(3544,\cdot)\)
\(\chi_{35525}(3609,\cdot)\)
\(\chi_{35525}(4314,\cdot)\)
\(\chi_{35525}(5429,\cdot)\)
\(\chi_{35525}(6269,\cdot)\)
\(\chi_{35525}(6414,\cdot)\)
\(\chi_{35525}(7214,\cdot)\)
\(\chi_{35525}(9139,\cdot)\)
\(\chi_{35525}(10714,\cdot)\)
\(\chi_{35525}(11029,\cdot)\)
\(\chi_{35525}(11419,\cdot)\)
\(\chi_{35525}(12534,\cdot)\)
\(\chi_{35525}(13204,\cdot)\)
\(\chi_{35525}(13519,\cdot)\)
\(\chi_{35525}(13904,\cdot)\)
\(\chi_{35525}(14319,\cdot)\)
\(\chi_{35525}(16244,\cdot)\)
\(\chi_{35525}(17404,\cdot)\)
\(\chi_{35525}(17754,\cdot)\)
\(\chi_{35525}(17819,\cdot)\)
\(\chi_{35525}(18134,\cdot)\)
\(\chi_{35525}(19639,\cdot)\)
\(\chi_{35525}(20309,\cdot)\)
\(\chi_{35525}(20479,\cdot)\)
\(\chi_{35525}(21009,\cdot)\)
\(\chi_{35525}(24509,\cdot)\)
\(\chi_{35525}(24859,\cdot)\)
...
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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 };
\((8527,28276,30626)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{2}{21}\right),e\left(\frac{9}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 35525 }(10714, a) \) |
\(1\) | \(1\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{71}{105}\right)\) |
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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)
