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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9577, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([676,545]))
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gp:[g,chi] = znchar(Mod(666, 9577))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9577.666");
| Modulus: | \(9577\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9577\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{9577}(77,\cdot)\)
\(\chi_{9577}(137,\cdot)\)
\(\chi_{9577}(178,\cdot)\)
\(\chi_{9577}(240,\cdot)\)
\(\chi_{9577}(320,\cdot)\)
\(\chi_{9577}(408,\cdot)\)
\(\chi_{9577}(544,\cdot)\)
\(\chi_{9577}(545,\cdot)\)
\(\chi_{9577}(666,\cdot)\)
\(\chi_{9577}(683,\cdot)\)
\(\chi_{9577}(713,\cdot)\)
\(\chi_{9577}(747,\cdot)\)
\(\chi_{9577}(809,\cdot)\)
\(\chi_{9577}(869,\cdot)\)
\(\chi_{9577}(879,\cdot)\)
\(\chi_{9577}(927,\cdot)\)
\(\chi_{9577}(937,\cdot)\)
\(\chi_{9577}(957,\cdot)\)
\(\chi_{9577}(1033,\cdot)\)
\(\chi_{9577}(1093,\cdot)\)
\(\chi_{9577}(1120,\cdot)\)
\(\chi_{9577}(1171,\cdot)\)
\(\chi_{9577}(1184,\cdot)\)
\(\chi_{9577}(1201,\cdot)\)
\(\chi_{9577}(1235,\cdot)\)
\(\chi_{9577}(1236,\cdot)\)
\(\chi_{9577}(1276,\cdot)\)
\(\chi_{9577}(1418,\cdot)\)
\(\chi_{9577}(1419,\cdot)\)
\(\chi_{9577}(1428,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((1100,5186)\) → \((e\left(\frac{13}{15}\right),e\left(\frac{109}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 9577 }(666, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{301}{780}\right)\) | \(e\left(\frac{193}{390}\right)\) | \(e\left(\frac{301}{390}\right)\) | \(e\left(\frac{199}{260}\right)\) | \(e\left(\frac{229}{260}\right)\) | \(e\left(\frac{139}{780}\right)\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{59}{390}\right)\) | \(e\left(\frac{22}{39}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
