from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,11,10]))
pari: [g,chi] = znchar(Mod(98,2475))
Basic properties
Modulus: | \(2475\) | |
Conductor: | \(825\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{825}(98,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.dg
\(\chi_{2475}(98,\cdot)\) \(\chi_{2475}(197,\cdot)\) \(\chi_{2475}(692,\cdot)\) \(\chi_{2475}(1088,\cdot)\) \(\chi_{2475}(1187,\cdot)\) \(\chi_{2475}(1583,\cdot)\) \(\chi_{2475}(2078,\cdot)\) \(\chi_{2475}(2177,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((551,2377,2026)\) → \((-1,e\left(\frac{11}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(98, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)