from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,24,15]))
pari: [g,chi] = znchar(Mod(281,5850))
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 \(\chi_{2925}(281,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.gm
\(\chi_{5850}(281,\cdot)\) \(\chi_{5850}(671,\cdot)\) \(\chi_{5850}(941,\cdot)\) \(\chi_{5850}(1721,\cdot)\) \(\chi_{5850}(1841,\cdot)\) \(\chi_{5850}(2111,\cdot)\) \(\chi_{5850}(2621,\cdot)\) \(\chi_{5850}(2891,\cdot)\) \(\chi_{5850}(3011,\cdot)\) \(\chi_{5850}(3281,\cdot)\) \(\chi_{5850}(3791,\cdot)\) \(\chi_{5850}(4061,\cdot)\) \(\chi_{5850}(4181,\cdot)\) \(\chi_{5850}(4961,\cdot)\) \(\chi_{5850}(5231,\cdot)\) \(\chi_{5850}(5621,\cdot)\)
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
\((3251,3277,2251)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(281, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)