from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,10,32,5]))
pari: [g,chi] = znchar(Mod(4667,5610))
Basic properties
Modulus: | \(5610\) | |
Conductor: | \(2805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2805}(1862,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5610.eq
\(\chi_{5610}(53,\cdot)\) \(\chi_{5610}(467,\cdot)\) \(\chi_{5610}(587,\cdot)\) \(\chi_{5610}(977,\cdot)\) \(\chi_{5610}(1103,\cdot)\) \(\chi_{5610}(2093,\cdot)\) \(\chi_{5610}(2117,\cdot)\) \(\chi_{5610}(2627,\cdot)\) \(\chi_{5610}(2633,\cdot)\) \(\chi_{5610}(3017,\cdot)\) \(\chi_{5610}(3623,\cdot)\) \(\chi_{5610}(4163,\cdot)\) \(\chi_{5610}(4547,\cdot)\) \(\chi_{5610}(4667,\cdot)\) \(\chi_{5610}(4673,\cdot)\) \(\chi_{5610}(5153,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((1871,3367,1531,3301)\) → \((-1,i,e\left(\frac{4}{5}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 5610 }(4667, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(1\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)