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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14014, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([205,63,175]))
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gp:[g,chi] = znchar(Mod(5717, 14014))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14014.5717");
| Modulus: | \(14014\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7007\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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 \(\chi_{7007}(5717,\cdot)\) |
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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);
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\(\chi_{14014}(17,\cdot)\)
\(\chi_{14014}(381,\cdot)\)
\(\chi_{14014}(563,\cdot)\)
\(\chi_{14014}(745,\cdot)\)
\(\chi_{14014}(985,\cdot)\)
\(\chi_{14014}(1349,\cdot)\)
\(\chi_{14014}(1531,\cdot)\)
\(\chi_{14014}(1713,\cdot)\)
\(\chi_{14014}(2019,\cdot)\)
\(\chi_{14014}(2565,\cdot)\)
\(\chi_{14014}(2747,\cdot)\)
\(\chi_{14014}(2987,\cdot)\)
\(\chi_{14014}(3533,\cdot)\)
\(\chi_{14014}(3715,\cdot)\)
\(\chi_{14014}(4021,\cdot)\)
\(\chi_{14014}(4385,\cdot)\)
\(\chi_{14014}(4567,\cdot)\)
\(\chi_{14014}(4749,\cdot)\)
\(\chi_{14014}(4989,\cdot)\)
\(\chi_{14014}(5353,\cdot)\)
\(\chi_{14014}(5535,\cdot)\)
\(\chi_{14014}(5717,\cdot)\)
\(\chi_{14014}(6023,\cdot)\)
\(\chi_{14014}(6387,\cdot)\)
\(\chi_{14014}(6569,\cdot)\)
\(\chi_{14014}(6751,\cdot)\)
\(\chi_{14014}(6991,\cdot)\)
\(\chi_{14014}(7355,\cdot)\)
\(\chi_{14014}(7537,\cdot)\)
\(\chi_{14014}(7719,\cdot)\)
...
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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 };
\((3433,6371,12937)\) → \((e\left(\frac{41}{42}\right),e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 14014 }(5717, a) \) |
\(1\) | \(1\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{1}{210}\right)\) |
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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)
