from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([8,27,18]))
pari: [g,chi] = znchar(Mod(94,5265))
Basic properties
Modulus: | \(5265\) | |
Conductor: | \(5265\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5265.gk
\(\chi_{5265}(94,\cdot)\) \(\chi_{5265}(529,\cdot)\) \(\chi_{5265}(679,\cdot)\) \(\chi_{5265}(1114,\cdot)\) \(\chi_{5265}(1264,\cdot)\) \(\chi_{5265}(1699,\cdot)\) \(\chi_{5265}(1849,\cdot)\) \(\chi_{5265}(2284,\cdot)\) \(\chi_{5265}(2434,\cdot)\) \(\chi_{5265}(2869,\cdot)\) \(\chi_{5265}(3019,\cdot)\) \(\chi_{5265}(3454,\cdot)\) \(\chi_{5265}(3604,\cdot)\) \(\chi_{5265}(4039,\cdot)\) \(\chi_{5265}(4189,\cdot)\) \(\chi_{5265}(4624,\cdot)\) \(\chi_{5265}(4774,\cdot)\) \(\chi_{5265}(5209,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((326,2107,2836)\) → \((e\left(\frac{4}{27}\right),-1,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 5265 }(94, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{54}\right)\) |
sage: chi.jacobi_sum(n)