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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(97344, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([156,117,0,136]))
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gp:[g,chi] = znchar(Mod(12151, 97344))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("97344.12151");
| Modulus: | \(97344\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5408\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(312\) |
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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 \(\chi_{5408}(3363,\cdot)\) |
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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: | no |
| 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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\(\chi_{97344}(55,\cdot)\)
\(\chi_{97344}(919,\cdot)\)
\(\chi_{97344}(1927,\cdot)\)
\(\chi_{97344}(2791,\cdot)\)
\(\chi_{97344}(3799,\cdot)\)
\(\chi_{97344}(4663,\cdot)\)
\(\chi_{97344}(5671,\cdot)\)
\(\chi_{97344}(6535,\cdot)\)
\(\chi_{97344}(7543,\cdot)\)
\(\chi_{97344}(8407,\cdot)\)
\(\chi_{97344}(9415,\cdot)\)
\(\chi_{97344}(10279,\cdot)\)
\(\chi_{97344}(11287,\cdot)\)
\(\chi_{97344}(12151,\cdot)\)
\(\chi_{97344}(14023,\cdot)\)
\(\chi_{97344}(15031,\cdot)\)
\(\chi_{97344}(15895,\cdot)\)
\(\chi_{97344}(16903,\cdot)\)
\(\chi_{97344}(18775,\cdot)\)
\(\chi_{97344}(19639,\cdot)\)
\(\chi_{97344}(20647,\cdot)\)
\(\chi_{97344}(21511,\cdot)\)
\(\chi_{97344}(22519,\cdot)\)
\(\chi_{97344}(23383,\cdot)\)
\(\chi_{97344}(24391,\cdot)\)
\(\chi_{97344}(25255,\cdot)\)
\(\chi_{97344}(26263,\cdot)\)
\(\chi_{97344}(27127,\cdot)\)
\(\chi_{97344}(28135,\cdot)\)
\(\chi_{97344}(28999,\cdot)\)
...
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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 };
\((45631,6085,43265,28225)\) → \((-1,e\left(\frac{3}{8}\right),1,e\left(\frac{17}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 97344 }(12151, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{104}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{85}{312}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{175}{312}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{59}{312}\right)\) |
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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)
