from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,33,5]))
pari: [g,chi] = znchar(Mod(223,5850))
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 \(\chi_{2925}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.gd
\(\chi_{5850}(223,\cdot)\) \(\chi_{5850}(787,\cdot)\) \(\chi_{5850}(817,\cdot)\) \(\chi_{5850}(1813,\cdot)\) \(\chi_{5850}(1987,\cdot)\) \(\chi_{5850}(2563,\cdot)\) \(\chi_{5850}(2983,\cdot)\) \(\chi_{5850}(3127,\cdot)\) \(\chi_{5850}(3733,\cdot)\) \(\chi_{5850}(4153,\cdot)\) \(\chi_{5850}(4297,\cdot)\) \(\chi_{5850}(4327,\cdot)\) \(\chi_{5850}(4903,\cdot)\) \(\chi_{5850}(5323,\cdot)\) \(\chi_{5850}(5467,\cdot)\) \(\chi_{5850}(5497,\cdot)\)
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
\((3251,3277,2251)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{20}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)