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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39715, base_ring=CyclotomicField(598))
M = H._module
chi = DirichletCharacter(H, M([299,161,507]))
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gp:[g,chi] = znchar(Mod(1299, 39715))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39715.1299");
| Modulus: | \(39715\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(39715\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(598\) |
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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);
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\(\chi_{39715}(129,\cdot)\)
\(\chi_{39715}(389,\cdot)\)
\(\chi_{39715}(454,\cdot)\)
\(\chi_{39715}(584,\cdot)\)
\(\chi_{39715}(649,\cdot)\)
\(\chi_{39715}(1039,\cdot)\)
\(\chi_{39715}(1104,\cdot)\)
\(\chi_{39715}(1169,\cdot)\)
\(\chi_{39715}(1299,\cdot)\)
\(\chi_{39715}(1429,\cdot)\)
\(\chi_{39715}(1624,\cdot)\)
\(\chi_{39715}(1754,\cdot)\)
\(\chi_{39715}(1819,\cdot)\)
\(\chi_{39715}(1949,\cdot)\)
\(\chi_{39715}(2014,\cdot)\)
\(\chi_{39715}(2079,\cdot)\)
\(\chi_{39715}(2144,\cdot)\)
\(\chi_{39715}(2859,\cdot)\)
\(\chi_{39715}(2924,\cdot)\)
\(\chi_{39715}(3184,\cdot)\)
\(\chi_{39715}(3444,\cdot)\)
\(\chi_{39715}(3509,\cdot)\)
\(\chi_{39715}(3639,\cdot)\)
\(\chi_{39715}(3704,\cdot)\)
\(\chi_{39715}(3899,\cdot)\)
\(\chi_{39715}(4094,\cdot)\)
\(\chi_{39715}(4159,\cdot)\)
\(\chi_{39715}(4354,\cdot)\)
\(\chi_{39715}(4484,\cdot)\)
\(\chi_{39715}(4679,\cdot)\)
...
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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 };
\((15887,24676,36336)\) → \((-1,e\left(\frac{7}{26}\right),e\left(\frac{39}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 39715 }(1299, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{299}\right)\) | \(e\left(\frac{503}{598}\right)\) | \(e\left(\frac{18}{299}\right)\) | \(e\left(\frac{521}{598}\right)\) | \(e\left(\frac{131}{299}\right)\) | \(e\left(\frac{27}{299}\right)\) | \(e\left(\frac{204}{299}\right)\) | \(e\left(\frac{199}{299}\right)\) | \(e\left(\frac{539}{598}\right)\) | \(e\left(\frac{140}{299}\right)\) |
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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)
