from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1520, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,27,20]))
pari: [g,chi] = znchar(Mod(1233,1520))
Basic properties
Modulus: | \(1520\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(93,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1520.dw
\(\chi_{1520}(17,\cdot)\) \(\chi_{1520}(177,\cdot)\) \(\chi_{1520}(593,\cdot)\) \(\chi_{1520}(833,\cdot)\) \(\chi_{1520}(897,\cdot)\) \(\chi_{1520}(993,\cdot)\) \(\chi_{1520}(1073,\cdot)\) \(\chi_{1520}(1137,\cdot)\) \(\chi_{1520}(1233,\cdot)\) \(\chi_{1520}(1297,\cdot)\) \(\chi_{1520}(1377,\cdot)\) \(\chi_{1520}(1393,\cdot)\)
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: | 36.0.619876750267203693326033178758188478035934269428253173828125.1 |
Values on generators
\((191,1141,1217,401)\) → \((1,1,-i,e\left(\frac{5}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 1520 }(1233, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)