from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,20,18]))
pari: [g,chi] = znchar(Mod(107,3850))
Basic properties
Modulus: | \(3850\) | |
Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{385}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.fz
\(\chi_{3850}(107,\cdot)\) \(\chi_{3850}(193,\cdot)\) \(\chi_{3850}(457,\cdot)\) \(\chi_{3850}(557,\cdot)\) \(\chi_{3850}(893,\cdot)\) \(\chi_{3850}(1157,\cdot)\) \(\chi_{3850}(1493,\cdot)\) \(\chi_{3850}(1843,\cdot)\) \(\chi_{3850}(1943,\cdot)\) \(\chi_{3850}(2207,\cdot)\) \(\chi_{3850}(2307,\cdot)\) \(\chi_{3850}(2543,\cdot)\) \(\chi_{3850}(2657,\cdot)\) \(\chi_{3850}(3357,\cdot)\) \(\chi_{3850}(3593,\cdot)\) \(\chi_{3850}(3693,\cdot)\)
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
\((2927,2201,1751)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3850 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) |
sage: chi.jacobi_sum(n)