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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17550, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([70,99,150]))
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gp:[g,chi] = znchar(Mod(7823, 17550))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17550.7823");
| Modulus: | \(17550\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8775\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{8775}(7823,\cdot)\) |
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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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\(\chi_{17550}(563,\cdot)\)
\(\chi_{17550}(797,\cdot)\)
\(\chi_{17550}(803,\cdot)\)
\(\chi_{17550}(1037,\cdot)\)
\(\chi_{17550}(1733,\cdot)\)
\(\chi_{17550}(1967,\cdot)\)
\(\chi_{17550}(1973,\cdot)\)
\(\chi_{17550}(2903,\cdot)\)
\(\chi_{17550}(3137,\cdot)\)
\(\chi_{17550}(3377,\cdot)\)
\(\chi_{17550}(4073,\cdot)\)
\(\chi_{17550}(4313,\cdot)\)
\(\chi_{17550}(4547,\cdot)\)
\(\chi_{17550}(5477,\cdot)\)
\(\chi_{17550}(5483,\cdot)\)
\(\chi_{17550}(5717,\cdot)\)
\(\chi_{17550}(6413,\cdot)\)
\(\chi_{17550}(6647,\cdot)\)
\(\chi_{17550}(6653,\cdot)\)
\(\chi_{17550}(6887,\cdot)\)
\(\chi_{17550}(7583,\cdot)\)
\(\chi_{17550}(7817,\cdot)\)
\(\chi_{17550}(7823,\cdot)\)
\(\chi_{17550}(8753,\cdot)\)
\(\chi_{17550}(8987,\cdot)\)
\(\chi_{17550}(9227,\cdot)\)
\(\chi_{17550}(9923,\cdot)\)
\(\chi_{17550}(10163,\cdot)\)
\(\chi_{17550}(10397,\cdot)\)
\(\chi_{17550}(11327,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((9101,9127,8101)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{11}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 17550 }(7823, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{17}{36}\right)\) |
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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)
