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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7434, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([145,145,54]))
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gp:[g,chi] = znchar(Mod(1895, 7434))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7434.1895");
| Modulus: | \(7434\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3717\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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 \(\chi_{3717}(1895,\cdot)\) |
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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);
|
\(\chi_{7434}(5,\cdot)\)
\(\chi_{7434}(257,\cdot)\)
\(\chi_{7434}(383,\cdot)\)
\(\chi_{7434}(479,\cdot)\)
\(\chi_{7434}(605,\cdot)\)
\(\chi_{7434}(635,\cdot)\)
\(\chi_{7434}(761,\cdot)\)
\(\chi_{7434}(1265,\cdot)\)
\(\chi_{7434}(1361,\cdot)\)
\(\chi_{7434}(1487,\cdot)\)
\(\chi_{7434}(1613,\cdot)\)
\(\chi_{7434}(1739,\cdot)\)
\(\chi_{7434}(1865,\cdot)\)
\(\chi_{7434}(1895,\cdot)\)
\(\chi_{7434}(2021,\cdot)\)
\(\chi_{7434}(2369,\cdot)\)
\(\chi_{7434}(2495,\cdot)\)
\(\chi_{7434}(2621,\cdot)\)
\(\chi_{7434}(2777,\cdot)\)
\(\chi_{7434}(2873,\cdot)\)
\(\chi_{7434}(2903,\cdot)\)
\(\chi_{7434}(2999,\cdot)\)
\(\chi_{7434}(3029,\cdot)\)
\(\chi_{7434}(3125,\cdot)\)
\(\chi_{7434}(3155,\cdot)\)
\(\chi_{7434}(3281,\cdot)\)
\(\chi_{7434}(3503,\cdot)\)
\(\chi_{7434}(3785,\cdot)\)
\(\chi_{7434}(3881,\cdot)\)
\(\chi_{7434}(3911,\cdot)\)
...
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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 };
\((3305,6373,4663)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{9}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7434 }(1895, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{87}\right)\) | \(e\left(\frac{161}{174}\right)\) | \(e\left(\frac{23}{174}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{167}{174}\right)\) | \(e\left(\frac{85}{174}\right)\) | \(e\left(\frac{34}{87}\right)\) | \(e\left(\frac{91}{174}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{64}{87}\right)\) |
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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)
