from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2850, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,32]))
pari: [g,chi] = znchar(Mod(2057,2850))
Basic properties
Modulus: | \(2850\) | |
Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{285}(62,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2850.cc
\(\chi_{2850}(443,\cdot)\) \(\chi_{2850}(557,\cdot)\) \(\chi_{2850}(593,\cdot)\) \(\chi_{2850}(707,\cdot)\) \(\chi_{2850}(1043,\cdot)\) \(\chi_{2850}(1157,\cdot)\) \(\chi_{2850}(1643,\cdot)\) \(\chi_{2850}(1757,\cdot)\) \(\chi_{2850}(1943,\cdot)\) \(\chi_{2850}(2057,\cdot)\) \(\chi_{2850}(2543,\cdot)\) \(\chi_{2850}(2657,\cdot)\)
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
\((1901,1027,1351)\) → \((-1,i,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2850 }(2057, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) |
sage: chi.jacobi_sum(n)