from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,18,11]))
pari: [g,chi] = znchar(Mod(3898,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{667}(563,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.cc
\(\chi_{6003}(244,\cdot)\) \(\chi_{6003}(766,\cdot)\) \(\chi_{6003}(1027,\cdot)\) \(\chi_{6003}(1351,\cdot)\) \(\chi_{6003}(1873,\cdot)\) \(\chi_{6003}(2593,\cdot)\) \(\chi_{6003}(2656,\cdot)\) \(\chi_{6003}(2917,\cdot)\) \(\chi_{6003}(3115,\cdot)\) \(\chi_{6003}(3700,\cdot)\) \(\chi_{6003}(3898,\cdot)\) \(\chi_{6003}(3961,\cdot)\) \(\chi_{6003}(4159,\cdot)\) \(\chi_{6003}(4483,\cdot)\) \(\chi_{6003}(4942,\cdot)\) \(\chi_{6003}(5005,\cdot)\) \(\chi_{6003}(5203,\cdot)\) \(\chi_{6003}(5527,\cdot)\) \(\chi_{6003}(5725,\cdot)\) \(\chi_{6003}(5788,\cdot)\)
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: | 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1 |
Values on generators
\((668,3133,4555)\) → \((1,e\left(\frac{9}{22}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(3898, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)