from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,11,6]))
pari: [g,chi] = znchar(Mod(217,8280))
Basic properties
| Modulus: | \(8280\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(102,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8280.fj
\(\chi_{8280}(217,\cdot)\) \(\chi_{8280}(433,\cdot)\) \(\chi_{8280}(793,\cdot)\) \(\chi_{8280}(937,\cdot)\) \(\chi_{8280}(1873,\cdot)\) \(\chi_{8280}(2593,\cdot)\) \(\chi_{8280}(3097,\cdot)\) \(\chi_{8280}(3457,\cdot)\) \(\chi_{8280}(4177,\cdot)\) \(\chi_{8280}(4753,\cdot)\) \(\chi_{8280}(4897,\cdot)\) \(\chi_{8280}(5113,\cdot)\) \(\chi_{8280}(5617,\cdot)\) \(\chi_{8280}(5833,\cdot)\) \(\chi_{8280}(5977,\cdot)\) \(\chi_{8280}(6553,\cdot)\) \(\chi_{8280}(7057,\cdot)\) \(\chi_{8280}(7273,\cdot)\) \(\chi_{8280}(7417,\cdot)\) \(\chi_{8280}(7633,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((2071,4141,4601,1657,3961)\) → \((1,1,1,i,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8280 }(217, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) |
sage: chi.jacobi_sum(n)