from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,0,18]))
pari: [g,chi] = znchar(Mod(4047,8470))
Basic properties
Modulus: | \(8470\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{605}(417,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.cb
\(\chi_{8470}(43,\cdot)\) \(\chi_{8470}(197,\cdot)\) \(\chi_{8470}(813,\cdot)\) \(\chi_{8470}(1583,\cdot)\) \(\chi_{8470}(1737,\cdot)\) \(\chi_{8470}(2353,\cdot)\) \(\chi_{8470}(2507,\cdot)\) \(\chi_{8470}(3123,\cdot)\) \(\chi_{8470}(3277,\cdot)\) \(\chi_{8470}(3893,\cdot)\) \(\chi_{8470}(4047,\cdot)\) \(\chi_{8470}(4663,\cdot)\) \(\chi_{8470}(4817,\cdot)\) \(\chi_{8470}(5433,\cdot)\) \(\chi_{8470}(5587,\cdot)\) \(\chi_{8470}(6203,\cdot)\) \(\chi_{8470}(6357,\cdot)\) \(\chi_{8470}(6973,\cdot)\) \(\chi_{8470}(7127,\cdot)\) \(\chi_{8470}(7897,\cdot)\)
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: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((6777,6051,7141)\) → \((i,1,e\left(\frac{9}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 8470 }(4047, a) \) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) |
sage: chi.jacobi_sum(n)