from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,0,39]))
pari: [g,chi] = znchar(Mod(113,5796))
Basic properties
Modulus: | \(5796\) | |
Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{207}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.fs
\(\chi_{5796}(113,\cdot)\) \(\chi_{5796}(281,\cdot)\) \(\chi_{5796}(365,\cdot)\) \(\chi_{5796}(617,\cdot)\) \(\chi_{5796}(1121,\cdot)\) \(\chi_{5796}(1625,\cdot)\) \(\chi_{5796}(1877,\cdot)\) \(\chi_{5796}(2045,\cdot)\) \(\chi_{5796}(2297,\cdot)\) \(\chi_{5796}(2549,\cdot)\) \(\chi_{5796}(2633,\cdot)\) \(\chi_{5796}(2885,\cdot)\) \(\chi_{5796}(3053,\cdot)\) \(\chi_{5796}(3557,\cdot)\) \(\chi_{5796}(3641,\cdot)\) \(\chi_{5796}(3809,\cdot)\) \(\chi_{5796}(4145,\cdot)\) \(\chi_{5796}(4565,\cdot)\) \(\chi_{5796}(4817,\cdot)\) \(\chi_{5796}(5573,\cdot)\)
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
\((2899,1289,829,4789)\) → \((1,e\left(\frac{5}{6}\right),1,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5796 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) |
sage: chi.jacobi_sum(n)