from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([20,27]))
pari: [g,chi] = znchar(Mod(73,810))
Basic properties
| Modulus: | \(810\) | |
| Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{135}(133,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 810.r
\(\chi_{810}(37,\cdot)\) \(\chi_{810}(73,\cdot)\) \(\chi_{810}(127,\cdot)\) \(\chi_{810}(253,\cdot)\) \(\chi_{810}(307,\cdot)\) \(\chi_{810}(343,\cdot)\) \(\chi_{810}(397,\cdot)\) \(\chi_{810}(523,\cdot)\) \(\chi_{810}(577,\cdot)\) \(\chi_{810}(613,\cdot)\) \(\chi_{810}(667,\cdot)\) \(\chi_{810}(793,\cdot)\)
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: | 36.0.7225377334561374804949923918873673793376691639423370361328125.1 |
Values on generators
\((731,487)\) → \((e\left(\frac{5}{9}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 810 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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)