from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5166, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,30,21]))
pari: [g,chi] = znchar(Mod(923,5166))
Basic properties
Modulus: | \(5166\) | |
Conductor: | \(2583\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2583}(923,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5166.fa
\(\chi_{5166}(923,\cdot)\) \(\chi_{5166}(1427,\cdot)\) \(\chi_{5166}(1553,\cdot)\) \(\chi_{5166}(1679,\cdot)\) \(\chi_{5166}(1847,\cdot)\) \(\chi_{5166}(1973,\cdot)\) \(\chi_{5166}(2099,\cdot)\) \(\chi_{5166}(2603,\cdot)\) \(\chi_{5166}(3569,\cdot)\) \(\chi_{5166}(3695,\cdot)\) \(\chi_{5166}(3821,\cdot)\) \(\chi_{5166}(4325,\cdot)\) \(\chi_{5166}(4367,\cdot)\) \(\chi_{5166}(4871,\cdot)\) \(\chi_{5166}(4997,\cdot)\) \(\chi_{5166}(5123,\cdot)\)
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
\((2297,2215,3655)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{7}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 5166 }(923, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)