from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,20,48]))
pari: [g,chi] = znchar(Mod(37,4592))
Basic properties
Modulus: | \(4592\) | |
Conductor: | \(4592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4592.hm
\(\chi_{4592}(37,\cdot)\) \(\chi_{4592}(221,\cdot)\) \(\chi_{4592}(877,\cdot)\) \(\chi_{4592}(1117,\cdot)\) \(\chi_{4592}(1453,\cdot)\) \(\chi_{4592}(1677,\cdot)\) \(\chi_{4592}(1773,\cdot)\) \(\chi_{4592}(2109,\cdot)\) \(\chi_{4592}(2333,\cdot)\) \(\chi_{4592}(2517,\cdot)\) \(\chi_{4592}(3173,\cdot)\) \(\chi_{4592}(3413,\cdot)\) \(\chi_{4592}(3749,\cdot)\) \(\chi_{4592}(3973,\cdot)\) \(\chi_{4592}(4069,\cdot)\) \(\chi_{4592}(4405,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((575,3445,3937,785)\) → \((1,i,e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 4592 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)