from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,55,36]))
pari: [g,chi] = znchar(Mod(271,5796))
Basic properties
| Modulus: | \(5796\) | |
| Conductor: | \(644\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{644}(271,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.eo
\(\chi_{5796}(271,\cdot)\) \(\chi_{5796}(703,\cdot)\) \(\chi_{5796}(775,\cdot)\) \(\chi_{5796}(955,\cdot)\) \(\chi_{5796}(1531,\cdot)\) \(\chi_{5796}(1711,\cdot)\) \(\chi_{5796}(1783,\cdot)\) \(\chi_{5796}(1963,\cdot)\) \(\chi_{5796}(2467,\cdot)\) \(\chi_{5796}(2539,\cdot)\) \(\chi_{5796}(2791,\cdot)\) \(\chi_{5796}(2971,\cdot)\) \(\chi_{5796}(3223,\cdot)\) \(\chi_{5796}(3295,\cdot)\) \(\chi_{5796}(3475,\cdot)\) \(\chi_{5796}(3799,\cdot)\) \(\chi_{5796}(4051,\cdot)\) \(\chi_{5796}(4303,\cdot)\) \(\chi_{5796}(5239,\cdot)\) \(\chi_{5796}(5743,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2899,1289,829,4789)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5796 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)