from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,48,35]))
pari: [g,chi] = znchar(Mod(11,1950))
Basic properties
Modulus: | \(1950\) | |
Conductor: | \(975\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{975}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1950.cw
\(\chi_{1950}(11,\cdot)\) \(\chi_{1950}(41,\cdot)\) \(\chi_{1950}(71,\cdot)\) \(\chi_{1950}(371,\cdot)\) \(\chi_{1950}(431,\cdot)\) \(\chi_{1950}(461,\cdot)\) \(\chi_{1950}(761,\cdot)\) \(\chi_{1950}(791,\cdot)\) \(\chi_{1950}(821,\cdot)\) \(\chi_{1950}(1181,\cdot)\) \(\chi_{1950}(1211,\cdot)\) \(\chi_{1950}(1241,\cdot)\) \(\chi_{1950}(1541,\cdot)\) \(\chi_{1950}(1571,\cdot)\) \(\chi_{1950}(1631,\cdot)\) \(\chi_{1950}(1931,\cdot)\)
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
\((1301,1327,301)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1950 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)