from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1870, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,16,35]))
pari: [g,chi] = znchar(Mod(291,1870))
Basic properties
Modulus: | \(1870\) | |
Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{187}(104,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1870.ck
\(\chi_{1870}(291,\cdot)\) \(\chi_{1870}(621,\cdot)\) \(\chi_{1870}(631,\cdot)\) \(\chi_{1870}(801,\cdot)\) \(\chi_{1870}(841,\cdot)\) \(\chi_{1870}(961,\cdot)\) \(\chi_{1870}(971,\cdot)\) \(\chi_{1870}(1131,\cdot)\) \(\chi_{1870}(1171,\cdot)\) \(\chi_{1870}(1181,\cdot)\) \(\chi_{1870}(1301,\cdot)\) \(\chi_{1870}(1351,\cdot)\) \(\chi_{1870}(1511,\cdot)\) \(\chi_{1870}(1521,\cdot)\) \(\chi_{1870}(1681,\cdot)\) \(\chi_{1870}(1851,\cdot)\)
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: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((1497,1531,1431)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{7}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1870 }(291, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |
sage: chi.jacobi_sum(n)