from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,42,55]))
pari: [g,chi] = znchar(Mod(59,5850))
Basic properties
Modulus: | \(5850\) | |
Conductor: | \(2925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2925}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.gr
\(\chi_{5850}(59,\cdot)\) \(\chi_{5850}(119,\cdot)\) \(\chi_{5850}(479,\cdot)\) \(\chi_{5850}(1229,\cdot)\) \(\chi_{5850}(1289,\cdot)\) \(\chi_{5850}(1319,\cdot)\) \(\chi_{5850}(2459,\cdot)\) \(\chi_{5850}(2489,\cdot)\) \(\chi_{5850}(2819,\cdot)\) \(\chi_{5850}(3569,\cdot)\) \(\chi_{5850}(3629,\cdot)\) \(\chi_{5850}(3659,\cdot)\) \(\chi_{5850}(3989,\cdot)\) \(\chi_{5850}(4739,\cdot)\) \(\chi_{5850}(4829,\cdot)\) \(\chi_{5850}(5159,\cdot)\)
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
\((3251,3277,2251)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5850 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)