from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2400, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,20,28]))
pari: [g,chi] = znchar(Mod(59,2400))
Basic properties
| Modulus: | \(2400\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2400.ed
\(\chi_{2400}(59,\cdot)\) \(\chi_{2400}(179,\cdot)\) \(\chi_{2400}(419,\cdot)\) \(\chi_{2400}(539,\cdot)\) \(\chi_{2400}(659,\cdot)\) \(\chi_{2400}(779,\cdot)\) \(\chi_{2400}(1019,\cdot)\) \(\chi_{2400}(1139,\cdot)\) \(\chi_{2400}(1259,\cdot)\) \(\chi_{2400}(1379,\cdot)\) \(\chi_{2400}(1619,\cdot)\) \(\chi_{2400}(1739,\cdot)\) \(\chi_{2400}(1859,\cdot)\) \(\chi_{2400}(1979,\cdot)\) \(\chi_{2400}(2219,\cdot)\) \(\chi_{2400}(2339,\cdot)\)
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: | 40.40.53955375229419889123391977410055372800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((1951,901,1601,577)\) → \((-1,e\left(\frac{1}{8}\right),-1,e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2400 }(59, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)