from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,12,15]))
pari: [g,chi] = znchar(Mod(603,4004))
Basic properties
Modulus: | \(4004\) | |
Conductor: | \(572\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{572}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4004.fr
\(\chi_{4004}(603,\cdot)\) \(\chi_{4004}(1191,\cdot)\) \(\chi_{4004}(1555,\cdot)\) \(\chi_{4004}(1919,\cdot)\) \(\chi_{4004}(2423,\cdot)\) \(\chi_{4004}(2787,\cdot)\) \(\chi_{4004}(3151,\cdot)\) \(\chi_{4004}(3375,\cdot)\)
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
\((2003,3433,365,925)\) → \((-1,1,e\left(\frac{3}{5}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
\( \chi_{ 4004 }(603, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)