from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,0,23]))
pari: [g,chi] = znchar(Mod(967,2001))
Basic properties
Modulus: | \(2001\) | |
Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{29}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.be
\(\chi_{2001}(346,\cdot)\) \(\chi_{2001}(553,\cdot)\) \(\chi_{2001}(967,\cdot)\) \(\chi_{2001}(1036,\cdot)\) \(\chi_{2001}(1105,\cdot)\) \(\chi_{2001}(1174,\cdot)\) \(\chi_{2001}(1381,\cdot)\) \(\chi_{2001}(1519,\cdot)\) \(\chi_{2001}(1726,\cdot)\) \(\chi_{2001}(1795,\cdot)\) \(\chi_{2001}(1864,\cdot)\) \(\chi_{2001}(1933,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((668,1132,553)\) → \((1,1,e\left(\frac{23}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2001 }(967, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)