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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,0,55,168]))
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gp:[g,chi] = znchar(Mod(119, 8712))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8712.119");
| Modulus: | \(8712\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4356\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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 \(\chi_{4356}(119,\cdot)\) |
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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: | even |
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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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\(\chi_{8712}(47,\cdot)\)
\(\chi_{8712}(119,\cdot)\)
\(\chi_{8712}(191,\cdot)\)
\(\chi_{8712}(311,\cdot)\)
\(\chi_{8712}(335,\cdot)\)
\(\chi_{8712}(383,\cdot)\)
\(\chi_{8712}(455,\cdot)\)
\(\chi_{8712}(599,\cdot)\)
\(\chi_{8712}(839,\cdot)\)
\(\chi_{8712}(911,\cdot)\)
\(\chi_{8712}(983,\cdot)\)
\(\chi_{8712}(1103,\cdot)\)
\(\chi_{8712}(1127,\cdot)\)
\(\chi_{8712}(1175,\cdot)\)
\(\chi_{8712}(1247,\cdot)\)
\(\chi_{8712}(1391,\cdot)\)
\(\chi_{8712}(1631,\cdot)\)
\(\chi_{8712}(1895,\cdot)\)
\(\chi_{8712}(1919,\cdot)\)
\(\chi_{8712}(1967,\cdot)\)
\(\chi_{8712}(2039,\cdot)\)
\(\chi_{8712}(2183,\cdot)\)
\(\chi_{8712}(2495,\cdot)\)
\(\chi_{8712}(2567,\cdot)\)
\(\chi_{8712}(2687,\cdot)\)
\(\chi_{8712}(2711,\cdot)\)
\(\chi_{8712}(2759,\cdot)\)
\(\chi_{8712}(2831,\cdot)\)
\(\chi_{8712}(2975,\cdot)\)
\(\chi_{8712}(3215,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{165})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 330 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((6535,4357,1937,5689)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{28}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(119, a) \) |
\(1\) | \(1\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{13}{55}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
