from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,9,7]))
pari: [g,chi] = znchar(Mod(757,8880))
Basic properties
Modulus: | \(8880\) | |
Conductor: | \(2960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2960}(757,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.pe
\(\chi_{8880}(757,\cdot)\) \(\chi_{8880}(997,\cdot)\) \(\chi_{8880}(2677,\cdot)\) \(\chi_{8880}(3613,\cdot)\) \(\chi_{8880}(5293,\cdot)\) \(\chi_{8880}(5533,\cdot)\) \(\chi_{8880}(6493,\cdot)\) \(\chi_{8880}(6517,\cdot)\) \(\chi_{8880}(7237,\cdot)\) \(\chi_{8880}(7933,\cdot)\) \(\chi_{8880}(8653,\cdot)\) \(\chi_{8880}(8677,\cdot)\)
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
\((5551,6661,5921,1777,8401)\) → \((1,i,1,i,e\left(\frac{7}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 8880 }(757, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{8}{9}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)