from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,42,55]))
pari: [g,chi] = znchar(Mod(59,2925))
Basic properties
Modulus: | \(2925\) | |
Conductor: | \(2925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 2925.he
\(\chi_{2925}(59,\cdot)\) \(\chi_{2925}(119,\cdot)\) \(\chi_{2925}(479,\cdot)\) \(\chi_{2925}(644,\cdot)\) \(\chi_{2925}(704,\cdot)\) \(\chi_{2925}(734,\cdot)\) \(\chi_{2925}(1064,\cdot)\) \(\chi_{2925}(1229,\cdot)\) \(\chi_{2925}(1289,\cdot)\) \(\chi_{2925}(1319,\cdot)\) \(\chi_{2925}(1814,\cdot)\) \(\chi_{2925}(1904,\cdot)\) \(\chi_{2925}(2234,\cdot)\) \(\chi_{2925}(2459,\cdot)\) \(\chi_{2925}(2489,\cdot)\) \(\chi_{2925}(2819,\cdot)\)
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
\((326,352,2251)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 2925 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)