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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10470, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([58,58,59]))
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gp:[g,chi] = znchar(Mod(4529, 10470))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10470.4529");
| Modulus: | \(10470\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5235\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(116\) |
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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 \(\chi_{5235}(4529,\cdot)\) |
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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: | 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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\(\chi_{10470}(179,\cdot)\)
\(\chi_{10470}(359,\cdot)\)
\(\chi_{10470}(659,\cdot)\)
\(\chi_{10470}(719,\cdot)\)
\(\chi_{10470}(989,\cdot)\)
\(\chi_{10470}(1019,\cdot)\)
\(\chi_{10470}(1229,\cdot)\)
\(\chi_{10470}(1349,\cdot)\)
\(\chi_{10470}(1499,\cdot)\)
\(\chi_{10470}(1529,\cdot)\)
\(\chi_{10470}(1559,\cdot)\)
\(\chi_{10470}(1739,\cdot)\)
\(\chi_{10470}(2129,\cdot)\)
\(\chi_{10470}(2159,\cdot)\)
\(\chi_{10470}(3089,\cdot)\)
\(\chi_{10470}(3149,\cdot)\)
\(\chi_{10470}(3179,\cdot)\)
\(\chi_{10470}(3239,\cdot)\)
\(\chi_{10470}(3359,\cdot)\)
\(\chi_{10470}(3389,\cdot)\)
\(\chi_{10470}(3479,\cdot)\)
\(\chi_{10470}(3569,\cdot)\)
\(\chi_{10470}(4109,\cdot)\)
\(\chi_{10470}(4199,\cdot)\)
\(\chi_{10470}(4289,\cdot)\)
\(\chi_{10470}(4319,\cdot)\)
\(\chi_{10470}(4439,\cdot)\)
\(\chi_{10470}(4499,\cdot)\)
\(\chi_{10470}(4529,\cdot)\)
\(\chi_{10470}(4589,\cdot)\)
...
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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: |
$\Q(\zeta_{116})$ |
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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 116 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3491,8377,3841)\) → \((-1,-1,e\left(\frac{59}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 10470 }(4529, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) |
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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)
