from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,44]))
pari: [g,chi] = znchar(Mod(293,3100))
Basic properties
Modulus: | \(3100\) | |
Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{155}(138,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3100.ga
\(\chi_{3100}(193,\cdot)\) \(\chi_{3100}(257,\cdot)\) \(\chi_{3100}(293,\cdot)\) \(\chi_{3100}(493,\cdot)\) \(\chi_{3100}(793,\cdot)\) \(\chi_{3100}(857,\cdot)\) \(\chi_{3100}(1157,\cdot)\) \(\chi_{3100}(1557,\cdot)\) \(\chi_{3100}(1657,\cdot)\) \(\chi_{3100}(1693,\cdot)\) \(\chi_{3100}(1857,\cdot)\) \(\chi_{3100}(1993,\cdot)\) \(\chi_{3100}(2157,\cdot)\) \(\chi_{3100}(2593,\cdot)\) \(\chi_{3100}(2893,\cdot)\) \(\chi_{3100}(3057,\cdot)\)
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
\((1551,2977,1801)\) → \((1,-i,e\left(\frac{11}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 3100 }(293, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)