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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,56,22]))
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gp:[g,chi] = znchar(Mod(683, 4002))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4002.683");
| Modulus: | \(4002\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2001\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(154\) |
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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: | no, induced from \(\chi_{2001}(683,\cdot)\) |
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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);
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\(\chi_{4002}(197,\cdot)\)
\(\chi_{4002}(239,\cdot)\)
\(\chi_{4002}(257,\cdot)\)
\(\chi_{4002}(335,\cdot)\)
\(\chi_{4002}(371,\cdot)\)
\(\chi_{4002}(455,\cdot)\)
\(\chi_{4002}(509,\cdot)\)
\(\chi_{4002}(545,\cdot)\)
\(\chi_{4002}(587,\cdot)\)
\(\chi_{4002}(629,\cdot)\)
\(\chi_{4002}(683,\cdot)\)
\(\chi_{4002}(719,\cdot)\)
\(\chi_{4002}(749,\cdot)\)
\(\chi_{4002}(761,\cdot)\)
\(\chi_{4002}(857,\cdot)\)
\(\chi_{4002}(923,\cdot)\)
\(\chi_{4002}(1067,\cdot)\)
\(\chi_{4002}(1097,\cdot)\)
\(\chi_{4002}(1205,\cdot)\)
\(\chi_{4002}(1271,\cdot)\)
\(\chi_{4002}(1283,\cdot)\)
\(\chi_{4002}(1301,\cdot)\)
\(\chi_{4002}(1415,\cdot)\)
\(\chi_{4002}(1457,\cdot)\)
\(\chi_{4002}(1475,\cdot)\)
\(\chi_{4002}(1499,\cdot)\)
\(\chi_{4002}(1553,\cdot)\)
\(\chi_{4002}(1589,\cdot)\)
\(\chi_{4002}(1619,\cdot)\)
\(\chi_{4002}(1649,\cdot)\)
...
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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 };
\((2669,3133,553)\) → \((-1,e\left(\frac{4}{11}\right),e\left(\frac{1}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 4002 }(683, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{53}{154}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{5}{77}\right)\) |
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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)
