from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,9,4]))
pari: [g,chi] = znchar(Mod(37,2475))
Basic properties
Modulus: | \(2475\) | |
Conductor: | \(275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{275}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.dt
\(\chi_{2475}(37,\cdot)\) \(\chi_{2475}(1153,\cdot)\) \(\chi_{2475}(1477,\cdot)\) \(\chi_{2475}(1567,\cdot)\) \(\chi_{2475}(1747,\cdot)\) \(\chi_{2475}(1873,\cdot)\) \(\chi_{2475}(2413,\cdot)\) \(\chi_{2475}(2458,\cdot)\)
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
\((551,2377,2026)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)