from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5166, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,50,33]))
pari: [g,chi] = znchar(Mod(5,5166))
Basic properties
| Modulus: | \(5166\) | |
| Conductor: | \(2583\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2583}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5166.fc
\(\chi_{5166}(5,\cdot)\) \(\chi_{5166}(131,\cdot)\) \(\chi_{5166}(635,\cdot)\) \(\chi_{5166}(1109,\cdot)\) \(\chi_{5166}(1235,\cdot)\) \(\chi_{5166}(1361,\cdot)\) \(\chi_{5166}(1865,\cdot)\) \(\chi_{5166}(2399,\cdot)\) \(\chi_{5166}(2903,\cdot)\) \(\chi_{5166}(3029,\cdot)\) \(\chi_{5166}(3155,\cdot)\) \(\chi_{5166}(3629,\cdot)\) \(\chi_{5166}(4133,\cdot)\) \(\chi_{5166}(4259,\cdot)\) \(\chi_{5166}(4385,\cdot)\) \(\chi_{5166}(5045,\cdot)\)
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
\((2297,2215,3655)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5166 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
sage: chi.jacobi_sum(n)