from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2850, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,34]))
pari: [g,chi] = znchar(Mod(1093,2850))
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 \(\chi_{95}(48,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2850.cb
\(\chi_{2850}(193,\cdot)\) \(\chi_{2850}(307,\cdot)\) \(\chi_{2850}(793,\cdot)\) \(\chi_{2850}(907,\cdot)\) \(\chi_{2850}(1093,\cdot)\) \(\chi_{2850}(1207,\cdot)\) \(\chi_{2850}(1693,\cdot)\) \(\chi_{2850}(1807,\cdot)\) \(\chi_{2850}(2143,\cdot)\) \(\chi_{2850}(2257,\cdot)\) \(\chi_{2850}(2293,\cdot)\) \(\chi_{2850}(2407,\cdot)\)
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: | \(\Q(\zeta_{95})^+\) |
Values on generators
\((1901,1027,1351)\) → \((1,-i,e\left(\frac{17}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2850 }(1093, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) |
sage: chi.jacobi_sum(n)