from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,25,20,24]))
pari: [g,chi] = znchar(Mod(71,4800))
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2400}(971,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.eo
\(\chi_{4800}(71,\cdot)\) \(\chi_{4800}(311,\cdot)\) \(\chi_{4800}(791,\cdot)\) \(\chi_{4800}(1031,\cdot)\) \(\chi_{4800}(1271,\cdot)\) \(\chi_{4800}(1511,\cdot)\) \(\chi_{4800}(1991,\cdot)\) \(\chi_{4800}(2231,\cdot)\) \(\chi_{4800}(2471,\cdot)\) \(\chi_{4800}(2711,\cdot)\) \(\chi_{4800}(3191,\cdot)\) \(\chi_{4800}(3431,\cdot)\) \(\chi_{4800}(3671,\cdot)\) \(\chi_{4800}(3911,\cdot)\) \(\chi_{4800}(4391,\cdot)\) \(\chi_{4800}(4631,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((4351,901,1601,577)\) → \((-1,e\left(\frac{5}{8}\right),-1,e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4800 }(71, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)