from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,15,32]))
pari: [g,chi] = znchar(Mod(1113,4030))
Basic properties
Modulus: | \(4030\) | |
Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2015}(1113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.gj
\(\chi_{4030}(317,\cdot)\) \(\chi_{4030}(577,\cdot)\) \(\chi_{4030}(723,\cdot)\) \(\chi_{4030}(1113,\cdot)\) \(\chi_{4030}(1227,\cdot)\) \(\chi_{4030}(1373,\cdot)\) \(\chi_{4030}(2153,\cdot)\) \(\chi_{4030}(2283,\cdot)\) \(\chi_{4030}(2397,\cdot)\) \(\chi_{4030}(2673,\cdot)\) \(\chi_{4030}(2787,\cdot)\) \(\chi_{4030}(2933,\cdot)\) \(\chi_{4030}(3047,\cdot)\) \(\chi_{4030}(3583,\cdot)\) \(\chi_{4030}(3827,\cdot)\) \(\chi_{4030}(3957,\cdot)\)
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
\((807,1861,2731)\) → \((-i,i,e\left(\frac{8}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(1113, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)