from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,27,23]))
pari: [g,chi] = znchar(Mod(1633,8880))
Basic properties
Modulus: | \(8880\) | |
Conductor: | \(185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{185}(153,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.qm
\(\chi_{8880}(1297,\cdot)\) \(\chi_{8880}(1537,\cdot)\) \(\chi_{8880}(1633,\cdot)\) \(\chi_{8880}(2353,\cdot)\) \(\chi_{8880}(2497,\cdot)\) \(\chi_{8880}(3793,\cdot)\) \(\chi_{8880}(3937,\cdot)\) \(\chi_{8880}(4657,\cdot)\) \(\chi_{8880}(4753,\cdot)\) \(\chi_{8880}(4993,\cdot)\) \(\chi_{8880}(6673,\cdot)\) \(\chi_{8880}(8497,\cdot)\)
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: | 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.1 |
Values on generators
\((5551,6661,5921,1777,8401)\) → \((1,1,1,-i,e\left(\frac{23}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 8880 }(1633, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{5}{18}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)