from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([2,27,0]))
pari: [g,chi] = znchar(Mod(218,4725))
Basic properties
| Modulus: | \(4725\) | |
| 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 \(\chi_{135}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.fe
\(\chi_{4725}(218,\cdot)\) \(\chi_{4725}(407,\cdot)\) \(\chi_{4725}(743,\cdot)\) \(\chi_{4725}(932,\cdot)\) \(\chi_{4725}(1793,\cdot)\) \(\chi_{4725}(1982,\cdot)\) \(\chi_{4725}(2318,\cdot)\) \(\chi_{4725}(2507,\cdot)\) \(\chi_{4725}(3368,\cdot)\) \(\chi_{4725}(3557,\cdot)\) \(\chi_{4725}(3893,\cdot)\) \(\chi_{4725}(4082,\cdot)\)
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: | \(\Q(\zeta_{135})^+\) |
Values on generators
\((4376,1702,2026)\) → \((e\left(\frac{1}{18}\right),-i,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 4725 }(218, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
sage: chi.jacobi_sum(n)