from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,23]))
pari: [g,chi] = znchar(Mod(3403,4002))
Basic properties
| Modulus: | \(4002\) | |
| Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{667}(68,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4002.bd
\(\chi_{4002}(229,\cdot)\) \(\chi_{4002}(367,\cdot)\) \(\chi_{4002}(781,\cdot)\) \(\chi_{4002}(1471,\cdot)\) \(\chi_{4002}(1609,\cdot)\) \(\chi_{4002}(2161,\cdot)\) \(\chi_{4002}(2299,\cdot)\) \(\chi_{4002}(2989,\cdot)\) \(\chi_{4002}(3403,\cdot)\) \(\chi_{4002}(3541,\cdot)\) \(\chi_{4002}(3817,\cdot)\) \(\chi_{4002}(3955,\cdot)\)
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: | 28.28.35394489068231220324814698212289719250778220848093751207381.1 |
Values on generators
\((2669,3133,553)\) → \((1,-1,e\left(\frac{23}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 4002 }(3403, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) |
sage: chi.jacobi_sum(n)