from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,2,0]))
pari: [g,chi] = znchar(Mod(85,2268))
Basic properties
| Modulus: | \(2268\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2268.cn
\(\chi_{2268}(85,\cdot)\) \(\chi_{2268}(169,\cdot)\) \(\chi_{2268}(337,\cdot)\) \(\chi_{2268}(421,\cdot)\) \(\chi_{2268}(589,\cdot)\) \(\chi_{2268}(673,\cdot)\) \(\chi_{2268}(841,\cdot)\) \(\chi_{2268}(925,\cdot)\) \(\chi_{2268}(1093,\cdot)\) \(\chi_{2268}(1177,\cdot)\) \(\chi_{2268}(1345,\cdot)\) \(\chi_{2268}(1429,\cdot)\) \(\chi_{2268}(1597,\cdot)\) \(\chi_{2268}(1681,\cdot)\) \(\chi_{2268}(1849,\cdot)\) \(\chi_{2268}(1933,\cdot)\) \(\chi_{2268}(2101,\cdot)\) \(\chi_{2268}(2185,\cdot)\)
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 27 polynomial |
Values on generators
\((1135,1541,325)\) → \((1,e\left(\frac{1}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(85, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)