from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,8]))
pari: [g,chi] = znchar(Mod(2869,4020))
Basic properties
| Modulus: | \(4020\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{335}(189,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.cz
\(\chi_{4020}(49,\cdot)\) \(\chi_{4020}(169,\cdot)\) \(\chi_{4020}(289,\cdot)\) \(\chi_{4020}(529,\cdot)\) \(\chi_{4020}(709,\cdot)\) \(\chi_{4020}(1009,\cdot)\) \(\chi_{4020}(1309,\cdot)\) \(\chi_{4020}(1729,\cdot)\) \(\chi_{4020}(1789,\cdot)\) \(\chi_{4020}(1909,\cdot)\) \(\chi_{4020}(1969,\cdot)\) \(\chi_{4020}(2029,\cdot)\) \(\chi_{4020}(2209,\cdot)\) \(\chi_{4020}(2569,\cdot)\) \(\chi_{4020}(2629,\cdot)\) \(\chi_{4020}(2869,\cdot)\) \(\chi_{4020}(3289,\cdot)\) \(\chi_{4020}(3769,\cdot)\) \(\chi_{4020}(3829,\cdot)\) \(\chi_{4020}(4009,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((1,1,-1,e\left(\frac{4}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(2869, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{14}{33}\right)\) |
sage: chi.jacobi_sum(n)