from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,0,19]))
pari: [g,chi] = znchar(Mod(461,2001))
Basic properties
Modulus: | \(2001\) | |
Conductor: | \(87\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{87}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.bc
\(\chi_{2001}(47,\cdot)\) \(\chi_{2001}(185,\cdot)\) \(\chi_{2001}(392,\cdot)\) \(\chi_{2001}(461,\cdot)\) \(\chi_{2001}(530,\cdot)\) \(\chi_{2001}(599,\cdot)\) \(\chi_{2001}(1013,\cdot)\) \(\chi_{2001}(1220,\cdot)\) \(\chi_{2001}(1634,\cdot)\) \(\chi_{2001}(1703,\cdot)\) \(\chi_{2001}(1772,\cdot)\) \(\chi_{2001}(1841,\cdot)\)
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: | \(\Q(\zeta_{87})^+\) |
Values on generators
\((668,1132,553)\) → \((-1,1,e\left(\frac{19}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2001 }(461, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)