from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,1]))
pari: [g,chi] = znchar(Mod(89,4730))
Basic properties
| Modulus: | \(4730\) | |
| Conductor: | \(215\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{215}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.ck
\(\chi_{4730}(89,\cdot)\) \(\chi_{4730}(749,\cdot)\) \(\chi_{4730}(1189,\cdot)\) \(\chi_{4730}(1409,\cdot)\) \(\chi_{4730}(1739,\cdot)\) \(\chi_{4730}(2069,\cdot)\) \(\chi_{4730}(2179,\cdot)\) \(\chi_{4730}(2399,\cdot)\) \(\chi_{4730}(2729,\cdot)\) \(\chi_{4730}(3169,\cdot)\) \(\chi_{4730}(3939,\cdot)\) \(\chi_{4730}(4269,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.4472772095648327544266441640449519840379754819764800031247757188817501068115234375.1 |
Values on generators
\((947,431,1981)\) → \((-1,1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4730 }(89, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) |
sage: chi.jacobi_sum(n)