from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243675, base_ring=CyclotomicField(3420))
M = H._module
chi = DirichletCharacter(H, M([2850,513,1780]))
pari: [g,chi] = znchar(Mod(233,243675))
Basic properties
| Modulus: | \(243675\) | |
| Conductor: | \(81225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81225}(54383,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 243675.bdt
\(\chi_{243675}(17,\cdot)\) \(\chi_{243675}(233,\cdot)\) \(\chi_{243675}(1277,\cdot)\) \(\chi_{243675}(1358,\cdot)\) \(\chi_{243675}(1412,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | $\Q(\zeta_{3420})$ |
| Fixed field: | Number field defined by a degree 3420 polynomial (not computed) |
Values on generators
\((36101,77977,129601)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{89}{171}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 243675 }(233, a) \) | \(1\) | \(1\) | \(e\left(\frac{1723}{3420}\right)\) | \(e\left(\frac{13}{1710}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{583}{1140}\right)\) | \(e\left(\frac{61}{190}\right)\) | \(e\left(\frac{2567}{3420}\right)\) | \(e\left(\frac{562}{855}\right)\) | \(e\left(\frac{13}{855}\right)\) | \(e\left(\frac{259}{3420}\right)\) | \(e\left(\frac{2821}{3420}\right)\) |
sage: chi.jacobi_sum(n)