from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,18,16]))
pari: [g,chi] = znchar(Mod(1187,2736))
Basic properties
Modulus: | \(2736\) | |
Conductor: | \(912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{912}(275,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2736.gv
\(\chi_{2736}(35,\cdot)\) \(\chi_{2736}(251,\cdot)\) \(\chi_{2736}(899,\cdot)\) \(\chi_{2736}(1043,\cdot)\) \(\chi_{2736}(1187,\cdot)\) \(\chi_{2736}(1259,\cdot)\) \(\chi_{2736}(1403,\cdot)\) \(\chi_{2736}(1619,\cdot)\) \(\chi_{2736}(2267,\cdot)\) \(\chi_{2736}(2411,\cdot)\) \(\chi_{2736}(2555,\cdot)\) \(\chi_{2736}(2627,\cdot)\)
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
\((1711,2053,1217,1009)\) → \((-1,-i,-1,e\left(\frac{4}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 2736 }(1187, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) |
sage: chi.jacobi_sum(n)