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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([133,110]))
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pari:[g,chi] = znchar(Mod(13,1225))
| Modulus: | \(1225\) | |
| Conductor: | \(1225\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(140\) |
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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)
|
\(\chi_{1225}(13,\cdot)\)
\(\chi_{1225}(27,\cdot)\)
\(\chi_{1225}(62,\cdot)\)
\(\chi_{1225}(83,\cdot)\)
\(\chi_{1225}(153,\cdot)\)
\(\chi_{1225}(167,\cdot)\)
\(\chi_{1225}(188,\cdot)\)
\(\chi_{1225}(202,\cdot)\)
\(\chi_{1225}(223,\cdot)\)
\(\chi_{1225}(237,\cdot)\)
\(\chi_{1225}(258,\cdot)\)
\(\chi_{1225}(272,\cdot)\)
\(\chi_{1225}(328,\cdot)\)
\(\chi_{1225}(363,\cdot)\)
\(\chi_{1225}(377,\cdot)\)
\(\chi_{1225}(398,\cdot)\)
\(\chi_{1225}(412,\cdot)\)
\(\chi_{1225}(433,\cdot)\)
\(\chi_{1225}(447,\cdot)\)
\(\chi_{1225}(503,\cdot)\)
\(\chi_{1225}(517,\cdot)\)
\(\chi_{1225}(552,\cdot)\)
\(\chi_{1225}(573,\cdot)\)
\(\chi_{1225}(608,\cdot)\)
\(\chi_{1225}(622,\cdot)\)
\(\chi_{1225}(678,\cdot)\)
\(\chi_{1225}(692,\cdot)\)
\(\chi_{1225}(713,\cdot)\)
\(\chi_{1225}(727,\cdot)\)
\(\chi_{1225}(748,\cdot)\)
...
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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 ]
\((1177,101)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) |
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sage:chi.jacobi_sum(n)
