from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,1]))
pari: [g,chi] = znchar(Mod(39,370))
Basic properties
Modulus: | \(370\) | |
Conductor: | \(185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{185}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 370.bb
\(\chi_{370}(19,\cdot)\) \(\chi_{370}(39,\cdot)\) \(\chi_{370}(59,\cdot)\) \(\chi_{370}(69,\cdot)\) \(\chi_{370}(79,\cdot)\) \(\chi_{370}(89,\cdot)\) \(\chi_{370}(109,\cdot)\) \(\chi_{370}(129,\cdot)\) \(\chi_{370}(209,\cdot)\) \(\chi_{370}(239,\cdot)\) \(\chi_{370}(279,\cdot)\) \(\chi_{370}(309,\cdot)\)
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: | 36.0.29411719834995153896864925426307140281034671856927417346954345703125.1 |
Values on generators
\((297,261)\) → \((-1,e\left(\frac{1}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 370 }(39, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)