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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11850, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([0,39,265]))
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gp:[g,chi] = znchar(Mod(1429, 11850))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11850.1429");
| Modulus: | \(11850\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1975\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(390\) |
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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 \(\chi_{1975}(1429,\cdot)\) |
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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: | 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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\(\chi_{11850}(109,\cdot)\)
\(\chi_{11850}(139,\cdot)\)
\(\chi_{11850}(319,\cdot)\)
\(\chi_{11850}(379,\cdot)\)
\(\chi_{11850}(469,\cdot)\)
\(\chi_{11850}(559,\cdot)\)
\(\chi_{11850}(619,\cdot)\)
\(\chi_{11850}(679,\cdot)\)
\(\chi_{11850}(709,\cdot)\)
\(\chi_{11850}(739,\cdot)\)
\(\chi_{11850}(829,\cdot)\)
\(\chi_{11850}(1159,\cdot)\)
\(\chi_{11850}(1219,\cdot)\)
\(\chi_{11850}(1339,\cdot)\)
\(\chi_{11850}(1429,\cdot)\)
\(\chi_{11850}(1459,\cdot)\)
\(\chi_{11850}(1609,\cdot)\)
\(\chi_{11850}(1639,\cdot)\)
\(\chi_{11850}(1729,\cdot)\)
\(\chi_{11850}(1939,\cdot)\)
\(\chi_{11850}(2029,\cdot)\)
\(\chi_{11850}(2089,\cdot)\)
\(\chi_{11850}(2359,\cdot)\)
\(\chi_{11850}(2479,\cdot)\)
\(\chi_{11850}(2509,\cdot)\)
\(\chi_{11850}(2689,\cdot)\)
\(\chi_{11850}(2839,\cdot)\)
\(\chi_{11850}(2929,\cdot)\)
\(\chi_{11850}(2989,\cdot)\)
\(\chi_{11850}(3079,\cdot)\)
...
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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 };
\((7901,11377,8851)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{53}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 11850 }(1429, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{1}{390}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{106}{195}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{263}{390}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{47}{130}\right)\) |
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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)
