from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,27,30]))
pari: [g,chi] = znchar(Mod(68,4725))
Basic properties
Modulus: | \(4725\) | |
Conductor: | \(945\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{945}(68,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.fi
\(\chi_{4725}(68,\cdot)\) \(\chi_{4725}(257,\cdot)\) \(\chi_{4725}(668,\cdot)\) \(\chi_{4725}(857,\cdot)\) \(\chi_{4725}(1643,\cdot)\) \(\chi_{4725}(1832,\cdot)\) \(\chi_{4725}(2243,\cdot)\) \(\chi_{4725}(2432,\cdot)\) \(\chi_{4725}(3218,\cdot)\) \(\chi_{4725}(3407,\cdot)\) \(\chi_{4725}(3818,\cdot)\) \(\chi_{4725}(4007,\cdot)\)
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: | 36.0.1465697146251004474650413649141163426679210854529430772252466557978101074695587158203125.1 |
Values on generators
\((4376,1702,2026)\) → \((e\left(\frac{17}{18}\right),-i,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 4725 }(68, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-i\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
sage: chi.jacobi_sum(n)