from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([22,27,27]))
pari: [g,chi] = znchar(Mod(3769,5265))
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.gn
\(\chi_{5265}(259,\cdot)\) \(\chi_{5265}(454,\cdot)\) \(\chi_{5265}(844,\cdot)\) \(\chi_{5265}(1039,\cdot)\) \(\chi_{5265}(1429,\cdot)\) \(\chi_{5265}(1624,\cdot)\) \(\chi_{5265}(2014,\cdot)\) \(\chi_{5265}(2209,\cdot)\) \(\chi_{5265}(2599,\cdot)\) \(\chi_{5265}(2794,\cdot)\) \(\chi_{5265}(3184,\cdot)\) \(\chi_{5265}(3379,\cdot)\) \(\chi_{5265}(3769,\cdot)\) \(\chi_{5265}(3964,\cdot)\) \(\chi_{5265}(4354,\cdot)\) \(\chi_{5265}(4549,\cdot)\) \(\chi_{5265}(4939,\cdot)\) \(\chi_{5265}(5134,\cdot)\)
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
\((326,2107,2836)\) → \((e\left(\frac{11}{27}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 5265 }(3769, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{54}\right)\) |
sage: chi.jacobi_sum(n)