from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,21,55]))
pari: [g,chi] = znchar(Mod(553,5850))
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 \(\chi_{2925}(553,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.hp
\(\chi_{5850}(553,\cdot)\) \(\chi_{5850}(583,\cdot)\) \(\chi_{5850}(877,\cdot)\) \(\chi_{5850}(1627,\cdot)\) \(\chi_{5850}(1723,\cdot)\) \(\chi_{5850}(1753,\cdot)\) \(\chi_{5850}(2047,\cdot)\) \(\chi_{5850}(2797,\cdot)\) \(\chi_{5850}(2923,\cdot)\) \(\chi_{5850}(3217,\cdot)\) \(\chi_{5850}(3967,\cdot)\) \(\chi_{5850}(4063,\cdot)\) \(\chi_{5850}(4387,\cdot)\) \(\chi_{5850}(5137,\cdot)\) \(\chi_{5850}(5233,\cdot)\) \(\chi_{5850}(5263,\cdot)\)
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
\((3251,3277,2251)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{20}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(553, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)