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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(355008, base_ring=CyclotomicField(1806))
M = H._module
chi = DirichletCharacter(H, M([903,903,903,20]))
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gp:[g,chi] = znchar(Mod(10463, 355008))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("355008.10463");
| 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: | \(44376\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1806\) |
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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 \(\chi_{44376}(32651,\cdot)\) |
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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);
|
\(\chi_{355008}(95,\cdot)\)
\(\chi_{355008}(1055,\cdot)\)
\(\chi_{355008}(1631,\cdot)\)
\(\chi_{355008}(1823,\cdot)\)
\(\chi_{355008}(2207,\cdot)\)
\(\chi_{355008}(2783,\cdot)\)
\(\chi_{355008}(3551,\cdot)\)
\(\chi_{355008}(5087,\cdot)\)
\(\chi_{355008}(5471,\cdot)\)
\(\chi_{355008}(6431,\cdot)\)
\(\chi_{355008}(7391,\cdot)\)
\(\chi_{355008}(7583,\cdot)\)
\(\chi_{355008}(8351,\cdot)\)
\(\chi_{355008}(9311,\cdot)\)
\(\chi_{355008}(9887,\cdot)\)
\(\chi_{355008}(10079,\cdot)\)
\(\chi_{355008}(10463,\cdot)\)
\(\chi_{355008}(11039,\cdot)\)
\(\chi_{355008}(11807,\cdot)\)
\(\chi_{355008}(13343,\cdot)\)
\(\chi_{355008}(13727,\cdot)\)
\(\chi_{355008}(14687,\cdot)\)
\(\chi_{355008}(15647,\cdot)\)
\(\chi_{355008}(15839,\cdot)\)
\(\chi_{355008}(16607,\cdot)\)
\(\chi_{355008}(17567,\cdot)\)
\(\chi_{355008}(18143,\cdot)\)
\(\chi_{355008}(18335,\cdot)\)
\(\chi_{355008}(18719,\cdot)\)
\(\chi_{355008}(19295,\cdot)\)
...
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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: |
$\Q(\zeta_{903})$ |
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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 1806 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((321727,66565,118337,85057)\) → \((-1,-1,-1,e\left(\frac{10}{903}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 355008 }(10463, a) \) |
\(1\) | \(1\) | \(e\left(\frac{124}{903}\right)\) | \(e\left(\frac{55}{258}\right)\) | \(e\left(\frac{389}{602}\right)\) | \(e\left(\frac{31}{1806}\right)\) | \(e\left(\frac{1621}{1806}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{643}{903}\right)\) | \(e\left(\frac{248}{903}\right)\) | \(e\left(\frac{410}{903}\right)\) | \(e\left(\frac{743}{1806}\right)\) |
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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)
