from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,20,12]))
pari: [g,chi] = znchar(Mod(89,4800))
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2400}(389,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.er
\(\chi_{4800}(89,\cdot)\) \(\chi_{4800}(329,\cdot)\) \(\chi_{4800}(569,\cdot)\) \(\chi_{4800}(809,\cdot)\) \(\chi_{4800}(1289,\cdot)\) \(\chi_{4800}(1529,\cdot)\) \(\chi_{4800}(1769,\cdot)\) \(\chi_{4800}(2009,\cdot)\) \(\chi_{4800}(2489,\cdot)\) \(\chi_{4800}(2729,\cdot)\) \(\chi_{4800}(2969,\cdot)\) \(\chi_{4800}(3209,\cdot)\) \(\chi_{4800}(3689,\cdot)\) \(\chi_{4800}(3929,\cdot)\) \(\chi_{4800}(4169,\cdot)\) \(\chi_{4800}(4409,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.53955375229419889123391977410055372800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((4351,901,1601,577)\) → \((1,e\left(\frac{1}{8}\right),-1,e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4800 }(89, a) \) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)