from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,18,9,10]))
pari: [g,chi] = znchar(Mod(617,8880))
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 \(\chi_{4440}(2837,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.mn
\(\chi_{8880}(617,\cdot)\) \(\chi_{8880}(953,\cdot)\) \(\chi_{8880}(1817,\cdot)\) \(\chi_{8880}(2297,\cdot)\) \(\chi_{8880}(2393,\cdot)\) \(\chi_{8880}(2537,\cdot)\) \(\chi_{8880}(3593,\cdot)\) \(\chi_{8880}(4073,\cdot)\) \(\chi_{8880}(4313,\cdot)\) \(\chi_{8880}(7097,\cdot)\) \(\chi_{8880}(8057,\cdot)\) \(\chi_{8880}(8873,\cdot)\)
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
\((5551,6661,5921,1777,8401)\) → \((1,-1,-1,i,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 8880 }(617, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{18}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)