from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,9,50]))
pari: [g,chi] = znchar(Mod(233,5550))
Basic properties
| Modulus: | \(5550\) | |
| Conductor: | \(2775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2775}(233,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5550.dt
\(\chi_{5550}(233,\cdot)\) \(\chi_{5550}(323,\cdot)\) \(\chi_{5550}(677,\cdot)\) \(\chi_{5550}(767,\cdot)\) \(\chi_{5550}(1433,\cdot)\) \(\chi_{5550}(1787,\cdot)\) \(\chi_{5550}(1877,\cdot)\) \(\chi_{5550}(2453,\cdot)\) \(\chi_{5550}(2897,\cdot)\) \(\chi_{5550}(2987,\cdot)\) \(\chi_{5550}(3563,\cdot)\) \(\chi_{5550}(3653,\cdot)\) \(\chi_{5550}(4097,\cdot)\) \(\chi_{5550}(4673,\cdot)\) \(\chi_{5550}(4763,\cdot)\) \(\chi_{5550}(5117,\cdot)\)
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
\((3701,1777,2851)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 5550 }(233, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)