from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,18,27,31]))
pari: [g,chi] = znchar(Mod(503,8880))
Basic properties
Modulus: | \(8880\) | |
Conductor: | \(4440\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4440}(2723,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.nc
\(\chi_{8880}(503,\cdot)\) \(\chi_{8880}(647,\cdot)\) \(\chi_{8880}(1943,\cdot)\) \(\chi_{8880}(2087,\cdot)\) \(\chi_{8880}(2807,\cdot)\) \(\chi_{8880}(2903,\cdot)\) \(\chi_{8880}(3143,\cdot)\) \(\chi_{8880}(4823,\cdot)\) \(\chi_{8880}(6647,\cdot)\) \(\chi_{8880}(8327,\cdot)\) \(\chi_{8880}(8567,\cdot)\) \(\chi_{8880}(8663,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((5551,6661,5921,1777,8401)\) → \((-1,-1,-1,-i,e\left(\frac{31}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 8880 }(503, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{2}{9}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)