from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,5]))
pari: [g,chi] = znchar(Mod(1540,2001))
Basic properties
Modulus: | \(2001\) | |
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 \(\chi_{667}(206,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.bd
\(\chi_{2001}(160,\cdot)\) \(\chi_{2001}(229,\cdot)\) \(\chi_{2001}(298,\cdot)\) \(\chi_{2001}(367,\cdot)\) \(\chi_{2001}(781,\cdot)\) \(\chi_{2001}(988,\cdot)\) \(\chi_{2001}(1402,\cdot)\) \(\chi_{2001}(1471,\cdot)\) \(\chi_{2001}(1540,\cdot)\) \(\chi_{2001}(1609,\cdot)\) \(\chi_{2001}(1816,\cdot)\) \(\chi_{2001}(1954,\cdot)\)
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
\((668,1132,553)\) → \((1,-1,e\left(\frac{5}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2001 }(1540, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)