from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,18,35]))
pari: [g,chi] = znchar(Mod(89,5850))
Basic properties
Modulus: | \(5850\) | |
Conductor: | \(975\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{975}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.gp
\(\chi_{5850}(89,\cdot)\) \(\chi_{5850}(539,\cdot)\) \(\chi_{5850}(1259,\cdot)\) \(\chi_{5850}(1619,\cdot)\) \(\chi_{5850}(1709,\cdot)\) \(\chi_{5850}(2069,\cdot)\) \(\chi_{5850}(2429,\cdot)\) \(\chi_{5850}(2789,\cdot)\) \(\chi_{5850}(2879,\cdot)\) \(\chi_{5850}(3239,\cdot)\) \(\chi_{5850}(3959,\cdot)\) \(\chi_{5850}(4409,\cdot)\) \(\chi_{5850}(4769,\cdot)\) \(\chi_{5850}(5129,\cdot)\) \(\chi_{5850}(5219,\cdot)\) \(\chi_{5850}(5579,\cdot)\)
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
\((3251,3277,2251)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5850 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)