from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,4,11]))
pari: [g,chi] = znchar(Mod(1520,2001))
Basic properties
Modulus: | \(2001\) | |
Conductor: | \(2001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.bg
\(\chi_{2001}(41,\cdot)\) \(\chi_{2001}(104,\cdot)\) \(\chi_{2001}(128,\cdot)\) \(\chi_{2001}(215,\cdot)\) \(\chi_{2001}(278,\cdot)\) \(\chi_{2001}(302,\cdot)\) \(\chi_{2001}(476,\cdot)\) \(\chi_{2001}(650,\cdot)\) \(\chi_{2001}(800,\cdot)\) \(\chi_{2001}(887,\cdot)\) \(\chi_{2001}(974,\cdot)\) \(\chi_{2001}(998,\cdot)\) \(\chi_{2001}(1061,\cdot)\) \(\chi_{2001}(1085,\cdot)\) \(\chi_{2001}(1235,\cdot)\) \(\chi_{2001}(1346,\cdot)\) \(\chi_{2001}(1409,\cdot)\) \(\chi_{2001}(1520,\cdot)\) \(\chi_{2001}(1757,\cdot)\) \(\chi_{2001}(1844,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((668,1132,553)\) → \((-1,e\left(\frac{1}{11}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2001 }(1520, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)