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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([0,27,10]))
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gp:[g,chi] = znchar(Mod(437, 1296))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1296.437");
| Modulus: | \(1296\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1296\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(108\) |
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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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\(\chi_{1296}(5,\cdot)\)
\(\chi_{1296}(29,\cdot)\)
\(\chi_{1296}(77,\cdot)\)
\(\chi_{1296}(101,\cdot)\)
\(\chi_{1296}(149,\cdot)\)
\(\chi_{1296}(173,\cdot)\)
\(\chi_{1296}(221,\cdot)\)
\(\chi_{1296}(245,\cdot)\)
\(\chi_{1296}(293,\cdot)\)
\(\chi_{1296}(317,\cdot)\)
\(\chi_{1296}(365,\cdot)\)
\(\chi_{1296}(389,\cdot)\)
\(\chi_{1296}(437,\cdot)\)
\(\chi_{1296}(461,\cdot)\)
\(\chi_{1296}(509,\cdot)\)
\(\chi_{1296}(533,\cdot)\)
\(\chi_{1296}(581,\cdot)\)
\(\chi_{1296}(605,\cdot)\)
\(\chi_{1296}(653,\cdot)\)
\(\chi_{1296}(677,\cdot)\)
\(\chi_{1296}(725,\cdot)\)
\(\chi_{1296}(749,\cdot)\)
\(\chi_{1296}(797,\cdot)\)
\(\chi_{1296}(821,\cdot)\)
\(\chi_{1296}(869,\cdot)\)
\(\chi_{1296}(893,\cdot)\)
\(\chi_{1296}(941,\cdot)\)
\(\chi_{1296}(965,\cdot)\)
\(\chi_{1296}(1013,\cdot)\)
\(\chi_{1296}(1037,\cdot)\)
...
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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: |
$\Q(\zeta_{108})$ |
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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 108 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1135,325,1217)\) → \((1,i,e\left(\frac{5}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1296 }(437, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{23}{27}\right)\) |
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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)
