from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7524, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,3,0,1]))
pari: [g,chi] = znchar(Mod(1541,7524))
Basic properties
| Modulus: | \(7524\) | |
| Conductor: | \(171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{171}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7524.fo
\(\chi_{7524}(1541,\cdot)\) \(\chi_{7524}(2597,\cdot)\) \(\chi_{7524}(2993,\cdot)\) \(\chi_{7524}(3917,\cdot)\) \(\chi_{7524}(7349,\cdot)\) \(\chi_{7524}(7481,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | Number field defined by a degree 18 polynomial |
Values on generators
\((3763,6689,4105,2377)\) → \((1,e\left(\frac{1}{6}\right),1,e\left(\frac{1}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 7524 }(1541, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)