from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,25]))
pari: [g,chi] = znchar(Mod(97,1850))
Basic properties
Modulus: | \(1850\) | |
Conductor: | \(925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{925}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1850.bx
\(\chi_{1850}(97,\cdot)\) \(\chi_{1850}(103,\cdot)\) \(\chi_{1850}(267,\cdot)\) \(\chi_{1850}(273,\cdot)\) \(\chi_{1850}(467,\cdot)\) \(\chi_{1850}(473,\cdot)\) \(\chi_{1850}(637,\cdot)\) \(\chi_{1850}(837,\cdot)\) \(\chi_{1850}(1013,\cdot)\) \(\chi_{1850}(1213,\cdot)\) \(\chi_{1850}(1377,\cdot)\) \(\chi_{1850}(1383,\cdot)\) \(\chi_{1850}(1577,\cdot)\) \(\chi_{1850}(1583,\cdot)\) \(\chi_{1850}(1747,\cdot)\) \(\chi_{1850}(1753,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1777,1001)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1850 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)