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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13431, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,32,0]))
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gp:[g,chi] = znchar(Mod(1148, 13431))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13431.1148");
| Modulus: | \(13431\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(363\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{363}(59,\cdot)\) |
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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: | odd |
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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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\(\chi_{13431}(38,\cdot)\)
\(\chi_{13431}(482,\cdot)\)
\(\chi_{13431}(1037,\cdot)\)
\(\chi_{13431}(1148,\cdot)\)
\(\chi_{13431}(1259,\cdot)\)
\(\chi_{13431}(2258,\cdot)\)
\(\chi_{13431}(2369,\cdot)\)
\(\chi_{13431}(2480,\cdot)\)
\(\chi_{13431}(2924,\cdot)\)
\(\chi_{13431}(3479,\cdot)\)
\(\chi_{13431}(3701,\cdot)\)
\(\chi_{13431}(4145,\cdot)\)
\(\chi_{13431}(4700,\cdot)\)
\(\chi_{13431}(4811,\cdot)\)
\(\chi_{13431}(4922,\cdot)\)
\(\chi_{13431}(5366,\cdot)\)
\(\chi_{13431}(5921,\cdot)\)
\(\chi_{13431}(6032,\cdot)\)
\(\chi_{13431}(6143,\cdot)\)
\(\chi_{13431}(6587,\cdot)\)
\(\chi_{13431}(7253,\cdot)\)
\(\chi_{13431}(7364,\cdot)\)
\(\chi_{13431}(7808,\cdot)\)
\(\chi_{13431}(8363,\cdot)\)
\(\chi_{13431}(8474,\cdot)\)
\(\chi_{13431}(8585,\cdot)\)
\(\chi_{13431}(9029,\cdot)\)
\(\chi_{13431}(9584,\cdot)\)
\(\chi_{13431}(9695,\cdot)\)
\(\chi_{13431}(9806,\cdot)\)
...
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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 };
\((4478,1333,7624)\) → \((-1,e\left(\frac{16}{55}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 13431 }(1148, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) |
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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)
