from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,3,0]))
pari: [g,chi] = znchar(Mod(677,5850))
Basic properties
| Modulus: | \(5850\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.gl
\(\chi_{5850}(677,\cdot)\) \(\chi_{5850}(833,\cdot)\) \(\chi_{5850}(1067,\cdot)\) \(\chi_{5850}(1613,\cdot)\) \(\chi_{5850}(1847,\cdot)\) \(\chi_{5850}(2003,\cdot)\) \(\chi_{5850}(2237,\cdot)\) \(\chi_{5850}(2783,\cdot)\) \(\chi_{5850}(3017,\cdot)\) \(\chi_{5850}(3173,\cdot)\) \(\chi_{5850}(3953,\cdot)\) \(\chi_{5850}(4187,\cdot)\) \(\chi_{5850}(4577,\cdot)\) \(\chi_{5850}(5123,\cdot)\) \(\chi_{5850}(5513,\cdot)\) \(\chi_{5850}(5747,\cdot)\)
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
\((3251,3277,2251)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(677, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)