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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61969, base_ring=CyclotomicField(270))
M = H._module
chi = DirichletCharacter(H, M([198,265]))
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gp:[g,chi] = znchar(Mod(16382, 61969))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61969.16382");
| Modulus: | \(61969\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(61969\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(270\) |
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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: | yes |
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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: | odd |
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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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\(\chi_{61969}(979,\cdot)\)
\(\chi_{61969}(3822,\cdot)\)
\(\chi_{61969}(4266,\cdot)\)
\(\chi_{61969}(4453,\cdot)\)
\(\chi_{61969}(5590,\cdot)\)
\(\chi_{61969}(9004,\cdot)\)
\(\chi_{61969}(9588,\cdot)\)
\(\chi_{61969}(10647,\cdot)\)
\(\chi_{61969}(11074,\cdot)\)
\(\chi_{61969}(11272,\cdot)\)
\(\chi_{61969}(12326,\cdot)\)
\(\chi_{61969}(12449,\cdot)\)
\(\chi_{61969}(15001,\cdot)\)
\(\chi_{61969}(15169,\cdot)\)
\(\chi_{61969}(16382,\cdot)\)
\(\chi_{61969}(16644,\cdot)\)
\(\chi_{61969}(18111,\cdot)\)
\(\chi_{61969}(18862,\cdot)\)
\(\chi_{61969}(19167,\cdot)\)
\(\chi_{61969}(21710,\cdot)\)
\(\chi_{61969}(21813,\cdot)\)
\(\chi_{61969}(22379,\cdot)\)
\(\chi_{61969}(24128,\cdot)\)
\(\chi_{61969}(24256,\cdot)\)
\(\chi_{61969}(24859,\cdot)\)
\(\chi_{61969}(25067,\cdot)\)
\(\chi_{61969}(25708,\cdot)\)
\(\chi_{61969}(26670,\cdot)\)
\(\chi_{61969}(27414,\cdot)\)
\(\chi_{61969}(27579,\cdot)\)
...
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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 };
\((53974,7999)\) → \((e\left(\frac{11}{15}\right),e\left(\frac{53}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 61969 }(16382, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{193}{270}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{2}{135}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
