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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([70,63,180]))
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gp:[g,chi] = znchar(Mod(4054, 33075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33075.4054");
| 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: | 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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\(\chi_{33075}(169,\cdot)\)
\(\chi_{33075}(484,\cdot)\)
\(\chi_{33075}(904,\cdot)\)
\(\chi_{33075}(1114,\cdot)\)
\(\chi_{33075}(1219,\cdot)\)
\(\chi_{33075}(1429,\cdot)\)
\(\chi_{33075}(1534,\cdot)\)
\(\chi_{33075}(1744,\cdot)\)
\(\chi_{33075}(2164,\cdot)\)
\(\chi_{33075}(2479,\cdot)\)
\(\chi_{33075}(2689,\cdot)\)
\(\chi_{33075}(3004,\cdot)\)
\(\chi_{33075}(3109,\cdot)\)
\(\chi_{33075}(3319,\cdot)\)
\(\chi_{33075}(3634,\cdot)\)
\(\chi_{33075}(3739,\cdot)\)
\(\chi_{33075}(4054,\cdot)\)
\(\chi_{33075}(4369,\cdot)\)
\(\chi_{33075}(4579,\cdot)\)
\(\chi_{33075}(4684,\cdot)\)
\(\chi_{33075}(4894,\cdot)\)
\(\chi_{33075}(5209,\cdot)\)
\(\chi_{33075}(5314,\cdot)\)
\(\chi_{33075}(5629,\cdot)\)
\(\chi_{33075}(5839,\cdot)\)
\(\chi_{33075}(5944,\cdot)\)
\(\chi_{33075}(6154,\cdot)\)
\(\chi_{33075}(6259,\cdot)\)
\(\chi_{33075}(6784,\cdot)\)
\(\chi_{33075}(6889,\cdot)\)
...
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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 };
\((23276,15877,20926)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{1}{10}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 33075 }(4054, a) \) |
\(1\) | \(1\) | \(e\left(\frac{403}{630}\right)\) | \(e\left(\frac{88}{315}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{149}{315}\right)\) | \(e\left(\frac{137}{630}\right)\) | \(e\left(\frac{176}{315}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{71}{630}\right)\) | \(e\left(\frac{113}{630}\right)\) |
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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)
