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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(355008, base_ring=CyclotomicField(602))
M = H._module
chi = DirichletCharacter(H, M([0,301,0,542]))
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gp:[g,chi] = znchar(Mod(97, 355008))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("355008.97");
| Modulus: | \(355008\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14792\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(602\) |
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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 \(\chi_{14792}(7493,\cdot)\) |
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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);
|
\(\chi_{355008}(97,\cdot)\)
\(\chi_{355008}(2209,\cdot)\)
\(\chi_{355008}(3745,\cdot)\)
\(\chi_{355008}(4321,\cdot)\)
\(\chi_{355008}(4513,\cdot)\)
\(\chi_{355008}(5281,\cdot)\)
\(\chi_{355008}(8353,\cdot)\)
\(\chi_{355008}(10465,\cdot)\)
\(\chi_{355008}(12001,\cdot)\)
\(\chi_{355008}(12577,\cdot)\)
\(\chi_{355008}(12769,\cdot)\)
\(\chi_{355008}(13537,\cdot)\)
\(\chi_{355008}(16609,\cdot)\)
\(\chi_{355008}(18721,\cdot)\)
\(\chi_{355008}(20257,\cdot)\)
\(\chi_{355008}(20833,\cdot)\)
\(\chi_{355008}(21025,\cdot)\)
\(\chi_{355008}(21793,\cdot)\)
\(\chi_{355008}(24865,\cdot)\)
\(\chi_{355008}(26977,\cdot)\)
\(\chi_{355008}(28513,\cdot)\)
\(\chi_{355008}(29089,\cdot)\)
\(\chi_{355008}(30049,\cdot)\)
\(\chi_{355008}(33121,\cdot)\)
\(\chi_{355008}(35233,\cdot)\)
\(\chi_{355008}(36769,\cdot)\)
\(\chi_{355008}(37345,\cdot)\)
\(\chi_{355008}(37537,\cdot)\)
\(\chi_{355008}(38305,\cdot)\)
\(\chi_{355008}(41377,\cdot)\)
...
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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: |
$\Q(\zeta_{301})$ |
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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 602 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((321727,66565,118337,85057)\) → \((1,-1,1,e\left(\frac{271}{301}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 355008 }(97, a) \) |
\(1\) | \(1\) | \(e\left(\frac{159}{602}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{111}{602}\right)\) | \(e\left(\frac{509}{602}\right)\) | \(e\left(\frac{127}{301}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{178}{301}\right)\) | \(e\left(\frac{159}{301}\right)\) | \(e\left(\frac{249}{602}\right)\) | \(e\left(\frac{240}{301}\right)\) |
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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)
