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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,26,11]))
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pari:[g,chi] = znchar(Mod(4391,8004))
| Modulus: | \(8004\) | |
| Conductor: | \(8004\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
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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)
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\(\chi_{8004}(191,\cdot)\)
\(\chi_{8004}(539,\cdot)\)
\(\chi_{8004}(563,\cdot)\)
\(\chi_{8004}(911,\cdot)\)
\(\chi_{8004}(1259,\cdot)\)
\(\chi_{8004}(1583,\cdot)\)
\(\chi_{8004}(1607,\cdot)\)
\(\chi_{8004}(2627,\cdot)\)
\(\chi_{8004}(3323,\cdot)\)
\(\chi_{8004}(3671,\cdot)\)
\(\chi_{8004}(3695,\cdot)\)
\(\chi_{8004}(4019,\cdot)\)
\(\chi_{8004}(4367,\cdot)\)
\(\chi_{8004}(4391,\cdot)\)
\(\chi_{8004}(5435,\cdot)\)
\(\chi_{8004}(5783,\cdot)\)
\(\chi_{8004}(6455,\cdot)\)
\(\chi_{8004}(6827,\cdot)\)
\(\chi_{8004}(7151,\cdot)\)
\(\chi_{8004}(7871,\cdot)\)
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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 ]
\((4003,2669,3133,553)\) → \((-1,-1,e\left(\frac{13}{22}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(4391, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) |
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sage:chi.jacobi_sum(n)
