from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2850, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,32]))
pari: [g,chi] = znchar(Mod(43,2850))
Basic properties
Modulus: | \(2850\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2850.ca
\(\chi_{2850}(43,\cdot)\) \(\chi_{2850}(157,\cdot)\) \(\chi_{2850}(643,\cdot)\) \(\chi_{2850}(757,\cdot)\) \(\chi_{2850}(1393,\cdot)\) \(\chi_{2850}(1507,\cdot)\) \(\chi_{2850}(1543,\cdot)\) \(\chi_{2850}(1657,\cdot)\) \(\chi_{2850}(1993,\cdot)\) \(\chi_{2850}(2107,\cdot)\) \(\chi_{2850}(2593,\cdot)\) \(\chi_{2850}(2707,\cdot)\)
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: | 36.0.619876750267203693326033178758188478035934269428253173828125.1 |
Values on generators
\((1901,1027,1351)\) → \((1,-i,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2850 }(43, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) |
sage: chi.jacobi_sum(n)