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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2420, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,55,52]))
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gp:[g,chi] = znchar(Mod(1709, 2420))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2420.1709");
| Modulus: | \(2420\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(605\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{605}(499,\cdot)\) |
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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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\(\chi_{2420}(49,\cdot)\)
\(\chi_{2420}(69,\cdot)\)
\(\chi_{2420}(169,\cdot)\)
\(\chi_{2420}(229,\cdot)\)
\(\chi_{2420}(289,\cdot)\)
\(\chi_{2420}(389,\cdot)\)
\(\chi_{2420}(449,\cdot)\)
\(\chi_{2420}(489,\cdot)\)
\(\chi_{2420}(509,\cdot)\)
\(\chi_{2420}(609,\cdot)\)
\(\chi_{2420}(669,\cdot)\)
\(\chi_{2420}(709,\cdot)\)
\(\chi_{2420}(829,\cdot)\)
\(\chi_{2420}(889,\cdot)\)
\(\chi_{2420}(929,\cdot)\)
\(\chi_{2420}(949,\cdot)\)
\(\chi_{2420}(1109,\cdot)\)
\(\chi_{2420}(1149,\cdot)\)
\(\chi_{2420}(1169,\cdot)\)
\(\chi_{2420}(1269,\cdot)\)
\(\chi_{2420}(1329,\cdot)\)
\(\chi_{2420}(1369,\cdot)\)
\(\chi_{2420}(1389,\cdot)\)
\(\chi_{2420}(1489,\cdot)\)
\(\chi_{2420}(1549,\cdot)\)
\(\chi_{2420}(1589,\cdot)\)
\(\chi_{2420}(1609,\cdot)\)
\(\chi_{2420}(1709,\cdot)\)
\(\chi_{2420}(1769,\cdot)\)
\(\chi_{2420}(1809,\cdot)\)
...
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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: |
$\Q(\zeta_{55})$ |
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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 110 polynomial (not computed) |
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sage:chi.fixed_field()
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\((1211,1937,2301)\) → \((1,-1,e\left(\frac{26}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2420 }(1709, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{55}\right)\) |
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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)
