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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45738, base_ring=CyclotomicField(990))
M = H._module
chi = DirichletCharacter(H, M([880,0,36]))
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gp:[g,chi] = znchar(Mod(5461, 45738))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45738.5461");
| Modulus: | \(45738\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3267\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(495\) |
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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 \(\chi_{3267}(2194,\cdot)\) |
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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);
|
\(\chi_{45738}(169,\cdot)\)
\(\chi_{45738}(295,\cdot)\)
\(\chi_{45738}(421,\cdot)\)
\(\chi_{45738}(841,\cdot)\)
\(\chi_{45738}(1093,\cdot)\)
\(\chi_{45738}(1303,\cdot)\)
\(\chi_{45738}(1345,\cdot)\)
\(\chi_{45738}(1555,\cdot)\)
\(\chi_{45738}(1681,\cdot)\)
\(\chi_{45738}(1807,\cdot)\)
\(\chi_{45738}(2227,\cdot)\)
\(\chi_{45738}(2479,\cdot)\)
\(\chi_{45738}(2605,\cdot)\)
\(\chi_{45738}(2731,\cdot)\)
\(\chi_{45738}(2941,\cdot)\)
\(\chi_{45738}(3067,\cdot)\)
\(\chi_{45738}(3193,\cdot)\)
\(\chi_{45738}(3613,\cdot)\)
\(\chi_{45738}(3865,\cdot)\)
\(\chi_{45738}(3991,\cdot)\)
\(\chi_{45738}(4075,\cdot)\)
\(\chi_{45738}(4327,\cdot)\)
\(\chi_{45738}(4453,\cdot)\)
\(\chi_{45738}(4579,\cdot)\)
\(\chi_{45738}(4999,\cdot)\)
\(\chi_{45738}(5251,\cdot)\)
\(\chi_{45738}(5377,\cdot)\)
\(\chi_{45738}(5461,\cdot)\)
\(\chi_{45738}(5503,\cdot)\)
\(\chi_{45738}(5713,\cdot)\)
...
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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: |
$\Q(\zeta_{495})$ |
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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 495 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((38963,19603,42715)\) → \((e\left(\frac{8}{9}\right),1,e\left(\frac{2}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 45738 }(5461, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{495}\right)\) | \(e\left(\frac{388}{495}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{134}{495}\right)\) | \(e\left(\frac{251}{495}\right)\) | \(e\left(\frac{448}{495}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{469}{495}\right)\) |
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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)
