from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,4,35]))
pari: [g,chi] = znchar(Mod(79,4100))
Basic properties
Modulus: | \(4100\) | |
Conductor: | \(4100\) | 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 4100.hd
\(\chi_{4100}(79,\cdot)\) \(\chi_{4100}(219,\cdot)\) \(\chi_{4100}(519,\cdot)\) \(\chi_{4100}(659,\cdot)\) \(\chi_{4100}(1039,\cdot)\) \(\chi_{4100}(1339,\cdot)\) \(\chi_{4100}(1479,\cdot)\) \(\chi_{4100}(1719,\cdot)\) \(\chi_{4100}(1859,\cdot)\) \(\chi_{4100}(2159,\cdot)\) \(\chi_{4100}(2539,\cdot)\) \(\chi_{4100}(2679,\cdot)\) \(\chi_{4100}(2979,\cdot)\) \(\chi_{4100}(3119,\cdot)\) \(\chi_{4100}(3359,\cdot)\) \(\chi_{4100}(3939,\cdot)\)
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
\((2051,1477,3901)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{7}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 4100 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) |
sage: chi.jacobi_sum(n)