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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17328, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([171,171,171,43]))
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gp:[g,chi] = znchar(Mod(2807, 17328))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17328.2807");
| Modulus: | \(17328\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8664\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(342\) |
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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 \(\chi_{8664}(7139,\cdot)\) |
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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);
|
\(\chi_{17328}(71,\cdot)\)
\(\chi_{17328}(167,\cdot)\)
\(\chi_{17328}(599,\cdot)\)
\(\chi_{17328}(743,\cdot)\)
\(\chi_{17328}(839,\cdot)\)
\(\chi_{17328}(887,\cdot)\)
\(\chi_{17328}(983,\cdot)\)
\(\chi_{17328}(1079,\cdot)\)
\(\chi_{17328}(1511,\cdot)\)
\(\chi_{17328}(1655,\cdot)\)
\(\chi_{17328}(1799,\cdot)\)
\(\chi_{17328}(1895,\cdot)\)
\(\chi_{17328}(1991,\cdot)\)
\(\chi_{17328}(2423,\cdot)\)
\(\chi_{17328}(2567,\cdot)\)
\(\chi_{17328}(2663,\cdot)\)
\(\chi_{17328}(2711,\cdot)\)
\(\chi_{17328}(2807,\cdot)\)
\(\chi_{17328}(2903,\cdot)\)
\(\chi_{17328}(3335,\cdot)\)
\(\chi_{17328}(3479,\cdot)\)
\(\chi_{17328}(3575,\cdot)\)
\(\chi_{17328}(3623,\cdot)\)
\(\chi_{17328}(3719,\cdot)\)
\(\chi_{17328}(3815,\cdot)\)
\(\chi_{17328}(4247,\cdot)\)
\(\chi_{17328}(4391,\cdot)\)
\(\chi_{17328}(4487,\cdot)\)
\(\chi_{17328}(4535,\cdot)\)
\(\chi_{17328}(4727,\cdot)\)
...
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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 };
\((10831,12997,5777,8305)\) → \((-1,-1,-1,e\left(\frac{43}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 17328 }(2807, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{47}{342}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{181}{342}\right)\) |
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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)
