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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11515, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,46,57]))
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gp:[g,chi] = znchar(Mod(6449, 11515))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11515.6449");
| Modulus: | \(11515\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1645\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{1645}(1514,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{11515}(214,\cdot)\)
\(\chi_{11515}(569,\cdot)\)
\(\chi_{11515}(814,\cdot)\)
\(\chi_{11515}(1194,\cdot)\)
\(\chi_{11515}(1304,\cdot)\)
\(\chi_{11515}(1439,\cdot)\)
\(\chi_{11515}(1549,\cdot)\)
\(\chi_{11515}(1684,\cdot)\)
\(\chi_{11515}(2419,\cdot)\)
\(\chi_{11515}(2529,\cdot)\)
\(\chi_{11515}(3019,\cdot)\)
\(\chi_{11515}(3154,\cdot)\)
\(\chi_{11515}(3399,\cdot)\)
\(\chi_{11515}(3509,\cdot)\)
\(\chi_{11515}(3754,\cdot)\)
\(\chi_{11515}(3889,\cdot)\)
\(\chi_{11515}(4134,\cdot)\)
\(\chi_{11515}(4979,\cdot)\)
\(\chi_{11515}(5114,\cdot)\)
\(\chi_{11515}(5604,\cdot)\)
\(\chi_{11515}(6094,\cdot)\)
\(\chi_{11515}(6339,\cdot)\)
\(\chi_{11515}(6449,\cdot)\)
\(\chi_{11515}(6694,\cdot)\)
\(\chi_{11515}(6939,\cdot)\)
\(\chi_{11515}(7184,\cdot)\)
\(\chi_{11515}(7564,\cdot)\)
\(\chi_{11515}(7674,\cdot)\)
\(\chi_{11515}(7919,\cdot)\)
\(\chi_{11515}(8164,\cdot)\)
...
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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 };
\((4607,9166,9311)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 11515 }(6449, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{138}\right)\) | \(e\left(\frac{13}{138}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{41}{138}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{28}{69}\right)\) |
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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)
