from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,0,10]))
pari: [g,chi] = znchar(Mod(139,2352))
Basic properties
| Modulus: | \(2352\) | |
| Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{784}(139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2352.cr
\(\chi_{2352}(139,\cdot)\) \(\chi_{2352}(307,\cdot)\) \(\chi_{2352}(475,\cdot)\) \(\chi_{2352}(643,\cdot)\) \(\chi_{2352}(811,\cdot)\) \(\chi_{2352}(1147,\cdot)\) \(\chi_{2352}(1315,\cdot)\) \(\chi_{2352}(1483,\cdot)\) \(\chi_{2352}(1651,\cdot)\) \(\chi_{2352}(1819,\cdot)\) \(\chi_{2352}(1987,\cdot)\) \(\chi_{2352}(2323,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.271776353216347717810469630450516372938858574109997048774397001728.1 |
Values on generators
\((1471,1765,785,2257)\) → \((-1,i,1,e\left(\frac{5}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2352 }(139, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) |
sage: chi.jacobi_sum(n)