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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(305760, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([0,105,84,84,104,70]))
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pari:[g,chi] = znchar(Mod(149,305760))
| Modulus: | \(305760\) | |
| Conductor: | \(305760\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(168\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{305760}(149,\cdot)\)
\(\chi_{305760}(2069,\cdot)\)
\(\chi_{305760}(4349,\cdot)\)
\(\chi_{305760}(18029,\cdot)\)
\(\chi_{305760}(21989,\cdot)\)
\(\chi_{305760}(23909,\cdot)\)
\(\chi_{305760}(26189,\cdot)\)
\(\chi_{305760}(39869,\cdot)\)
\(\chi_{305760}(43829,\cdot)\)
\(\chi_{305760}(45749,\cdot)\)
\(\chi_{305760}(48029,\cdot)\)
\(\chi_{305760}(65669,\cdot)\)
\(\chi_{305760}(69869,\cdot)\)
\(\chi_{305760}(83549,\cdot)\)
\(\chi_{305760}(87509,\cdot)\)
\(\chi_{305760}(89429,\cdot)\)
\(\chi_{305760}(105389,\cdot)\)
\(\chi_{305760}(111269,\cdot)\)
\(\chi_{305760}(113549,\cdot)\)
\(\chi_{305760}(127229,\cdot)\)
\(\chi_{305760}(131189,\cdot)\)
\(\chi_{305760}(133109,\cdot)\)
\(\chi_{305760}(135389,\cdot)\)
\(\chi_{305760}(149069,\cdot)\)
\(\chi_{305760}(153029,\cdot)\)
\(\chi_{305760}(154949,\cdot)\)
\(\chi_{305760}(157229,\cdot)\)
\(\chi_{305760}(170909,\cdot)\)
\(\chi_{305760}(174869,\cdot)\)
\(\chi_{305760}(176789,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((95551,114661,101921,183457,18721,211681)\) → \((1,e\left(\frac{5}{8}\right),-1,-1,e\left(\frac{13}{21}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 305760 }(149, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) |
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sage:chi.jacobi_sum(n)
