from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,4]))
pari: [g,chi] = znchar(Mod(301,4400))
Basic properties
Modulus: | \(4400\) | |
Conductor: | \(176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{176}(125,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4400.je
\(\chi_{4400}(301,\cdot)\) \(\chi_{4400}(1301,\cdot)\) \(\chi_{4400}(1501,\cdot)\) \(\chi_{4400}(1901,\cdot)\) \(\chi_{4400}(2501,\cdot)\) \(\chi_{4400}(3501,\cdot)\) \(\chi_{4400}(3701,\cdot)\) \(\chi_{4400}(4101,\cdot)\)
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: | 20.20.1655513490330868290261743826894848.1 |
Values on generators
\((2751,3301,177,1201)\) → \((1,-i,1,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4400 }(301, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)