from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,33,36]))
pari: [g,chi] = znchar(Mod(41,5796))
Basic properties
| Modulus: | \(5796\) | |
| Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1449}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.fe
\(\chi_{5796}(41,\cdot)\) \(\chi_{5796}(209,\cdot)\) \(\chi_{5796}(545,\cdot)\) \(\chi_{5796}(1301,\cdot)\) \(\chi_{5796}(1553,\cdot)\) \(\chi_{5796}(1973,\cdot)\) \(\chi_{5796}(2309,\cdot)\) \(\chi_{5796}(2477,\cdot)\) \(\chi_{5796}(2561,\cdot)\) \(\chi_{5796}(3065,\cdot)\) \(\chi_{5796}(3233,\cdot)\) \(\chi_{5796}(3485,\cdot)\) \(\chi_{5796}(3569,\cdot)\) \(\chi_{5796}(3821,\cdot)\) \(\chi_{5796}(4073,\cdot)\) \(\chi_{5796}(4241,\cdot)\) \(\chi_{5796}(4493,\cdot)\) \(\chi_{5796}(4997,\cdot)\) \(\chi_{5796}(5501,\cdot)\) \(\chi_{5796}(5753,\cdot)\)
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
\((2899,1289,829,4789)\) → \((1,e\left(\frac{5}{6}\right),-1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5796 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) |
sage: chi.jacobi_sum(n)