from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,44,33,12]))
pari: [g,chi] = znchar(Mod(349,8280))
Basic properties
| Modulus: | \(8280\) | |
| Conductor: | \(8280\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 8280.fv
\(\chi_{8280}(349,\cdot)\) \(\chi_{8280}(949,\cdot)\) \(\chi_{8280}(1429,\cdot)\) \(\chi_{8280}(1669,\cdot)\) \(\chi_{8280}(1789,\cdot)\) \(\chi_{8280}(2509,\cdot)\) \(\chi_{8280}(2749,\cdot)\) \(\chi_{8280}(3109,\cdot)\) \(\chi_{8280}(3229,\cdot)\) \(\chi_{8280}(3949,\cdot)\) \(\chi_{8280}(4189,\cdot)\) \(\chi_{8280}(4309,\cdot)\) \(\chi_{8280}(4549,\cdot)\) \(\chi_{8280}(5269,\cdot)\) \(\chi_{8280}(5989,\cdot)\) \(\chi_{8280}(6469,\cdot)\) \(\chi_{8280}(6709,\cdot)\) \(\chi_{8280}(7069,\cdot)\) \(\chi_{8280}(7189,\cdot)\) \(\chi_{8280}(8269,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2071,4141,4601,1657,3961)\) → \((1,-1,e\left(\frac{2}{3}\right),-1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8280 }(349, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) |
sage: chi.jacobi_sum(n)