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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,21,32]))
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pari:[g,chi] = znchar(Mod(184,2205))
| Modulus: | \(2205\) | |
| Conductor: | \(2205\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2205}(184,\cdot)\)
\(\chi_{2205}(499,\cdot)\)
\(\chi_{2205}(529,\cdot)\)
\(\chi_{2205}(844,\cdot)\)
\(\chi_{2205}(1129,\cdot)\)
\(\chi_{2205}(1159,\cdot)\)
\(\chi_{2205}(1444,\cdot)\)
\(\chi_{2205}(1474,\cdot)\)
\(\chi_{2205}(1759,\cdot)\)
\(\chi_{2205}(1789,\cdot)\)
\(\chi_{2205}(2074,\cdot)\)
\(\chi_{2205}(2104,\cdot)\)
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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 ]
\((1226,442,1081)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{16}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(184, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) |
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sage:chi.jacobi_sum(n)
