from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,0,3]))
pari: [g,chi] = znchar(Mod(41,1755))
Basic properties
| Modulus: | \(1755\) | |
| Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{351}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1755.fe
\(\chi_{1755}(41,\cdot)\) \(\chi_{1755}(371,\cdot)\) \(\chi_{1755}(401,\cdot)\) \(\chi_{1755}(461,\cdot)\) \(\chi_{1755}(626,\cdot)\) \(\chi_{1755}(956,\cdot)\) \(\chi_{1755}(986,\cdot)\) \(\chi_{1755}(1046,\cdot)\) \(\chi_{1755}(1211,\cdot)\) \(\chi_{1755}(1541,\cdot)\) \(\chi_{1755}(1571,\cdot)\) \(\chi_{1755}(1631,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((326,352,1081)\) → \((e\left(\frac{17}{18}\right),1,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 1755 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)