from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,24,11]))
pari: [g,chi] = znchar(Mod(2042,6003))
Basic properties
| Modulus: | \(6003\) | |
| Conductor: | \(2001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2001}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.cb
\(\chi_{6003}(215,\cdot)\) \(\chi_{6003}(278,\cdot)\) \(\chi_{6003}(476,\cdot)\) \(\chi_{6003}(800,\cdot)\) \(\chi_{6003}(998,\cdot)\) \(\chi_{6003}(1061,\cdot)\) \(\chi_{6003}(1520,\cdot)\) \(\chi_{6003}(1844,\cdot)\) \(\chi_{6003}(2042,\cdot)\) \(\chi_{6003}(2105,\cdot)\) \(\chi_{6003}(2303,\cdot)\) \(\chi_{6003}(2888,\cdot)\) \(\chi_{6003}(3086,\cdot)\) \(\chi_{6003}(3347,\cdot)\) \(\chi_{6003}(3410,\cdot)\) \(\chi_{6003}(4130,\cdot)\) \(\chi_{6003}(4652,\cdot)\) \(\chi_{6003}(4976,\cdot)\) \(\chi_{6003}(5237,\cdot)\) \(\chi_{6003}(5759,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((668,3133,4555)\) → \((-1,e\left(\frac{6}{11}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6003 }(2042, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)