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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56550, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([70,84,0,45]))
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gp:[g,chi] = znchar(Mod(14171, 56550))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56550.14171");
| Modulus: | \(56550\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2175\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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 \(\chi_{2175}(1121,\cdot)\) |
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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);
|
\(\chi_{56550}(131,\cdot)\)
\(\chi_{56550}(2861,\cdot)\)
\(\chi_{56550}(4811,\cdot)\)
\(\chi_{56550}(7931,\cdot)\)
\(\chi_{56550}(8321,\cdot)\)
\(\chi_{56550}(8711,\cdot)\)
\(\chi_{56550}(9491,\cdot)\)
\(\chi_{56550}(9881,\cdot)\)
\(\chi_{56550}(10661,\cdot)\)
\(\chi_{56550}(11441,\cdot)\)
\(\chi_{56550}(14171,\cdot)\)
\(\chi_{56550}(14561,\cdot)\)
\(\chi_{56550}(16121,\cdot)\)
\(\chi_{56550}(16511,\cdot)\)
\(\chi_{56550}(19241,\cdot)\)
\(\chi_{56550}(19631,\cdot)\)
\(\chi_{56550}(20021,\cdot)\)
\(\chi_{56550}(21191,\cdot)\)
\(\chi_{56550}(21971,\cdot)\)
\(\chi_{56550}(22361,\cdot)\)
\(\chi_{56550}(25481,\cdot)\)
\(\chi_{56550}(25871,\cdot)\)
\(\chi_{56550}(27431,\cdot)\)
\(\chi_{56550}(27821,\cdot)\)
\(\chi_{56550}(30941,\cdot)\)
\(\chi_{56550}(31331,\cdot)\)
\(\chi_{56550}(32111,\cdot)\)
\(\chi_{56550}(33281,\cdot)\)
\(\chi_{56550}(33671,\cdot)\)
\(\chi_{56550}(34061,\cdot)\)
...
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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 };
\((18851,52027,21751,48751)\) → \((-1,e\left(\frac{3}{5}\right),1,e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 56550 }(14171, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{33}{140}\right)\) |
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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)
