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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,22,12]))
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pari:[g,chi] = znchar(Mod(1112,2415))
| Modulus: | \(2415\) | |
| Conductor: | \(2415\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2415}(62,\cdot)\)
\(\chi_{2415}(167,\cdot)\)
\(\chi_{2415}(188,\cdot)\)
\(\chi_{2415}(377,\cdot)\)
\(\chi_{2415}(587,\cdot)\)
\(\chi_{2415}(692,\cdot)\)
\(\chi_{2415}(818,\cdot)\)
\(\chi_{2415}(923,\cdot)\)
\(\chi_{2415}(1007,\cdot)\)
\(\chi_{2415}(1028,\cdot)\)
\(\chi_{2415}(1112,\cdot)\)
\(\chi_{2415}(1133,\cdot)\)
\(\chi_{2415}(1343,\cdot)\)
\(\chi_{2415}(1553,\cdot)\)
\(\chi_{2415}(1637,\cdot)\)
\(\chi_{2415}(1658,\cdot)\)
\(\chi_{2415}(1973,\cdot)\)
\(\chi_{2415}(2078,\cdot)\)
\(\chi_{2415}(2267,\cdot)\)
\(\chi_{2415}(2372,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,i,-1,e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(1112, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) |
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sage:chi.jacobi_sum(n)
