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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36850, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([231,66,10]))
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gp:[g,chi] = znchar(Mod(11059, 36850))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36850.11059");
| Modulus: | \(36850\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18425\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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 \(\chi_{18425}(11059,\cdot)\) |
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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);
|
\(\chi_{36850}(289,\cdot)\)
\(\chi_{36850}(609,\cdot)\)
\(\chi_{36850}(839,\cdot)\)
\(\chi_{36850}(1329,\cdot)\)
\(\chi_{36850}(1389,\cdot)\)
\(\chi_{36850}(2429,\cdot)\)
\(\chi_{36850}(2489,\cdot)\)
\(\chi_{36850}(3019,\cdot)\)
\(\chi_{36850}(3909,\cdot)\)
\(\chi_{36850}(4629,\cdot)\)
\(\chi_{36850}(5219,\cdot)\)
\(\chi_{36850}(5559,\cdot)\)
\(\chi_{36850}(6319,\cdot)\)
\(\chi_{36850}(6659,\cdot)\)
\(\chi_{36850}(6869,\cdot)\)
\(\chi_{36850}(6889,\cdot)\)
\(\chi_{36850}(7419,\cdot)\)
\(\chi_{36850}(7759,\cdot)\)
\(\chi_{36850}(7929,\cdot)\)
\(\chi_{36850}(7989,\cdot)\)
\(\chi_{36850}(8519,\cdot)\)
\(\chi_{36850}(9579,\cdot)\)
\(\chi_{36850}(10679,\cdot)\)
\(\chi_{36850}(10739,\cdot)\)
\(\chi_{36850}(11059,\cdot)\)
\(\chi_{36850}(11289,\cdot)\)
\(\chi_{36850}(11779,\cdot)\)
\(\chi_{36850}(11839,\cdot)\)
\(\chi_{36850}(12919,\cdot)\)
\(\chi_{36850}(13259,\cdot)\)
...
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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 };
\((35377,6701,13201)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{5}\right),e\left(\frac{1}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 36850 }(11059, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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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)
