from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,15,0,38]))
pari: [g,chi] = znchar(Mod(4663,4800))
Basic properties
Modulus: | \(4800\) | |
Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{800}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.ev
\(\chi_{4800}(487,\cdot)\) \(\chi_{4800}(823,\cdot)\) \(\chi_{4800}(967,\cdot)\) \(\chi_{4800}(1303,\cdot)\) \(\chi_{4800}(1447,\cdot)\) \(\chi_{4800}(1783,\cdot)\) \(\chi_{4800}(1927,\cdot)\) \(\chi_{4800}(2263,\cdot)\) \(\chi_{4800}(2887,\cdot)\) \(\chi_{4800}(3223,\cdot)\) \(\chi_{4800}(3367,\cdot)\) \(\chi_{4800}(3703,\cdot)\) \(\chi_{4800}(3847,\cdot)\) \(\chi_{4800}(4183,\cdot)\) \(\chi_{4800}(4327,\cdot)\) \(\chi_{4800}(4663,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((4351,901,1601,577)\) → \((-1,e\left(\frac{3}{8}\right),1,e\left(\frac{19}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4800 }(4663, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)