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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(396))
M = H._module
chi = DirichletCharacter(H, M([0,297,242,306]))
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gp:[g,chi] = znchar(Mod(42305, 836352))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.42305");
| Modulus: | \(836352\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(52272\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(396\) |
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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 \(\chi_{52272}(3101,\cdot)\) |
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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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\(\chi_{836352}(65,\cdot)\)
\(\chi_{836352}(4289,\cdot)\)
\(\chi_{836352}(12737,\cdot)\)
\(\chi_{836352}(16961,\cdot)\)
\(\chi_{836352}(29633,\cdot)\)
\(\chi_{836352}(38081,\cdot)\)
\(\chi_{836352}(42305,\cdot)\)
\(\chi_{836352}(50753,\cdot)\)
\(\chi_{836352}(54977,\cdot)\)
\(\chi_{836352}(63425,\cdot)\)
\(\chi_{836352}(67649,\cdot)\)
\(\chi_{836352}(76097,\cdot)\)
\(\chi_{836352}(80321,\cdot)\)
\(\chi_{836352}(88769,\cdot)\)
\(\chi_{836352}(92993,\cdot)\)
\(\chi_{836352}(101441,\cdot)\)
\(\chi_{836352}(105665,\cdot)\)
\(\chi_{836352}(114113,\cdot)\)
\(\chi_{836352}(126785,\cdot)\)
\(\chi_{836352}(131009,\cdot)\)
\(\chi_{836352}(139457,\cdot)\)
\(\chi_{836352}(143681,\cdot)\)
\(\chi_{836352}(152129,\cdot)\)
\(\chi_{836352}(156353,\cdot)\)
\(\chi_{836352}(169025,\cdot)\)
\(\chi_{836352}(177473,\cdot)\)
\(\chi_{836352}(181697,\cdot)\)
\(\chi_{836352}(190145,\cdot)\)
\(\chi_{836352}(194369,\cdot)\)
\(\chi_{836352}(202817,\cdot)\)
...
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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: |
$\Q(\zeta_{396})$ |
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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 396 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((137215,561925,123905,781057)\) → \((1,-i,e\left(\frac{11}{18}\right),e\left(\frac{17}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(42305, a) \) |
\(1\) | \(1\) | \(e\left(\frac{391}{396}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{73}{396}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{395}{396}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{89}{132}\right)\) |
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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)
