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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33075, base_ring=CyclotomicField(630))
M = H._module
chi = DirichletCharacter(H, M([35,63,510]))
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gp:[g,chi] = znchar(Mod(5429, 33075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33075.5429");
| Modulus: | \(33075\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(33075\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(630\) |
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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);
|
\(\chi_{33075}(389,\cdot)\)
\(\chi_{33075}(464,\cdot)\)
\(\chi_{33075}(779,\cdot)\)
\(\chi_{33075}(1019,\cdot)\)
\(\chi_{33075}(1094,\cdot)\)
\(\chi_{33075}(1334,\cdot)\)
\(\chi_{33075}(1409,\cdot)\)
\(\chi_{33075}(1964,\cdot)\)
\(\chi_{33075}(2279,\cdot)\)
\(\chi_{33075}(2354,\cdot)\)
\(\chi_{33075}(2594,\cdot)\)
\(\chi_{33075}(2669,\cdot)\)
\(\chi_{33075}(2984,\cdot)\)
\(\chi_{33075}(3539,\cdot)\)
\(\chi_{33075}(3614,\cdot)\)
\(\chi_{33075}(3854,\cdot)\)
\(\chi_{33075}(3929,\cdot)\)
\(\chi_{33075}(4169,\cdot)\)
\(\chi_{33075}(4484,\cdot)\)
\(\chi_{33075}(4559,\cdot)\)
\(\chi_{33075}(5189,\cdot)\)
\(\chi_{33075}(5429,\cdot)\)
\(\chi_{33075}(5504,\cdot)\)
\(\chi_{33075}(5744,\cdot)\)
\(\chi_{33075}(5819,\cdot)\)
\(\chi_{33075}(6059,\cdot)\)
\(\chi_{33075}(6134,\cdot)\)
\(\chi_{33075}(6689,\cdot)\)
\(\chi_{33075}(6764,\cdot)\)
\(\chi_{33075}(7004,\cdot)\)
...
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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 };
\((23276,15877,20926)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{1}{10}\right),e\left(\frac{17}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 33075 }(5429, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{64}{315}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{443}{630}\right)\) | \(e\left(\frac{37}{630}\right)\) | \(e\left(\frac{256}{315}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{571}{630}\right)\) | \(e\left(\frac{149}{315}\right)\) |
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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)
