from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,26]))
pari: [g,chi] = znchar(Mod(77,740))
Basic properties
| Modulus: | \(740\) | |
| 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 \(\chi_{185}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 740.by
\(\chi_{740}(77,\cdot)\) \(\chi_{740}(173,\cdot)\) \(\chi_{740}(213,\cdot)\) \(\chi_{740}(317,\cdot)\) \(\chi_{740}(337,\cdot)\) \(\chi_{740}(373,\cdot)\) \(\chi_{740}(437,\cdot)\) \(\chi_{740}(613,\cdot)\) \(\chi_{740}(617,\cdot)\) \(\chi_{740}(633,\cdot)\) \(\chi_{740}(657,\cdot)\) \(\chi_{740}(733,\cdot)\)
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: | 36.0.1552561224397970539319305607385300901659347120825982756912708282470703125.1 |
Values on generators
\((371,297,261)\) → \((1,i,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(77, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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)