from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1334, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,1]))
pari: [g,chi] = znchar(Mod(321,1334))
Basic properties
Modulus: | \(1334\) | |
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 \(\chi_{667}(321,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1334.o
\(\chi_{1334}(137,\cdot)\) \(\chi_{1334}(229,\cdot)\) \(\chi_{1334}(275,\cdot)\) \(\chi_{1334}(321,\cdot)\) \(\chi_{1334}(367,\cdot)\) \(\chi_{1334}(735,\cdot)\) \(\chi_{1334}(781,\cdot)\) \(\chi_{1334}(827,\cdot)\) \(\chi_{1334}(873,\cdot)\) \(\chi_{1334}(965,\cdot)\) \(\chi_{1334}(1149,\cdot)\) \(\chi_{1334}(1287,\cdot)\)
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
\((465,553)\) → \((-1,e\left(\frac{1}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 1334 }(321, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) |
sage: chi.jacobi_sum(n)