from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,3,20]))
pari: [g,chi] = znchar(Mod(1927,3150))
Basic properties
Modulus: | \(3150\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.ek
\(\chi_{3150}(37,\cdot)\) \(\chi_{3150}(163,\cdot)\) \(\chi_{3150}(487,\cdot)\) \(\chi_{3150}(613,\cdot)\) \(\chi_{3150}(667,\cdot)\) \(\chi_{3150}(1117,\cdot)\) \(\chi_{3150}(1297,\cdot)\) \(\chi_{3150}(1423,\cdot)\) \(\chi_{3150}(1747,\cdot)\) \(\chi_{3150}(1873,\cdot)\) \(\chi_{3150}(1927,\cdot)\) \(\chi_{3150}(2053,\cdot)\) \(\chi_{3150}(2377,\cdot)\) \(\chi_{3150}(2503,\cdot)\) \(\chi_{3150}(2683,\cdot)\) \(\chi_{3150}(3133,\cdot)\)
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
\((2801,127,451)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3150 }(1927, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)