from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,25,12,20]))
pari: [g,chi] = znchar(Mod(139,5600))
Basic properties
| Modulus: | \(5600\) | |
| Conductor: | \(5600\) | 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 5600.gy
\(\chi_{5600}(139,\cdot)\) \(\chi_{5600}(419,\cdot)\) \(\chi_{5600}(979,\cdot)\) \(\chi_{5600}(1259,\cdot)\) \(\chi_{5600}(1539,\cdot)\) \(\chi_{5600}(1819,\cdot)\) \(\chi_{5600}(2379,\cdot)\) \(\chi_{5600}(2659,\cdot)\) \(\chi_{5600}(2939,\cdot)\) \(\chi_{5600}(3219,\cdot)\) \(\chi_{5600}(3779,\cdot)\) \(\chi_{5600}(4059,\cdot)\) \(\chi_{5600}(4339,\cdot)\) \(\chi_{5600}(4619,\cdot)\) \(\chi_{5600}(5179,\cdot)\) \(\chi_{5600}(5459,\cdot)\)
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
\((351,4901,5377,801)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{3}{10}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 5600 }(139, a) \) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)