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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15950, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([7,42,15]))
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gp:[g,chi] = znchar(Mod(11229, 15950))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15950.11229");
| Modulus: | \(15950\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7975\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(70\) |
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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: | no, induced from \(\chi_{7975}(3254,\cdot)\) |
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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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\(\chi_{15950}(1369,\cdot)\)
\(\chi_{15950}(1919,\cdot)\)
\(\chi_{15950}(2429,\cdot)\)
\(\chi_{15950}(2469,\cdot)\)
\(\chi_{15950}(3589,\cdot)\)
\(\chi_{15950}(4459,\cdot)\)
\(\chi_{15950}(7379,\cdot)\)
\(\chi_{15950}(8519,\cdot)\)
\(\chi_{15950}(8539,\cdot)\)
\(\chi_{15950}(9409,\cdot)\)
\(\chi_{15950}(9579,\cdot)\)
\(\chi_{15950}(10739,\cdot)\)
\(\chi_{15950}(11229,\cdot)\)
\(\chi_{15950}(11609,\cdot)\)
\(\chi_{15950}(11779,\cdot)\)
\(\chi_{15950}(12329,\cdot)\)
\(\chi_{15950}(12389,\cdot)\)
\(\chi_{15950}(12939,\cdot)\)
\(\chi_{15950}(13259,\cdot)\)
\(\chi_{15950}(13469,\cdot)\)
\(\chi_{15950}(13489,\cdot)\)
\(\chi_{15950}(13809,\cdot)\)
\(\chi_{15950}(14359,\cdot)\)
\(\chi_{15950}(15669,\cdot)\)
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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 };
\((1277,7251,15401)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3}{5}\right),e\left(\frac{3}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(31\) |
| \( \chi_{ 15950 }(11229, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{43}{70}\right)\) |
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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)
