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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([50,50,28,25]))
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gp:[g,chi] = znchar(Mod(4831, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.4831");
| Modulus: | \(104000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(100\) |
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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 \(\chi_{13000}(11331,\cdot)\) |
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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);
|
\(\chi_{104000}(31,\cdot)\)
\(\chi_{104000}(671,\cdot)\)
\(\chi_{104000}(4191,\cdot)\)
\(\chi_{104000}(4831,\cdot)\)
\(\chi_{104000}(8991,\cdot)\)
\(\chi_{104000}(12511,\cdot)\)
\(\chi_{104000}(16671,\cdot)\)
\(\chi_{104000}(17311,\cdot)\)
\(\chi_{104000}(20831,\cdot)\)
\(\chi_{104000}(21471,\cdot)\)
\(\chi_{104000}(24991,\cdot)\)
\(\chi_{104000}(25631,\cdot)\)
\(\chi_{104000}(29791,\cdot)\)
\(\chi_{104000}(33311,\cdot)\)
\(\chi_{104000}(37471,\cdot)\)
\(\chi_{104000}(38111,\cdot)\)
\(\chi_{104000}(41631,\cdot)\)
\(\chi_{104000}(42271,\cdot)\)
\(\chi_{104000}(45791,\cdot)\)
\(\chi_{104000}(46431,\cdot)\)
\(\chi_{104000}(50591,\cdot)\)
\(\chi_{104000}(54111,\cdot)\)
\(\chi_{104000}(58271,\cdot)\)
\(\chi_{104000}(58911,\cdot)\)
\(\chi_{104000}(62431,\cdot)\)
\(\chi_{104000}(63071,\cdot)\)
\(\chi_{104000}(66591,\cdot)\)
\(\chi_{104000}(67231,\cdot)\)
\(\chi_{104000}(71391,\cdot)\)
\(\chi_{104000}(74911,\cdot)\)
...
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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 };
\((74751,58501,77377,64001)\) → \((-1,-1,e\left(\frac{7}{25}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(4831, a) \) |
\(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) |
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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)
