from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6900, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,30]))
pari: [g,chi] = znchar(Mod(157,6900))
Basic properties
| Modulus: | \(6900\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(42,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6900.cq
\(\chi_{6900}(157,\cdot)\) \(\chi_{6900}(457,\cdot)\) \(\chi_{6900}(493,\cdot)\) \(\chi_{6900}(757,\cdot)\) \(\chi_{6900}(793,\cdot)\) \(\chi_{6900}(1693,\cdot)\) \(\chi_{6900}(1993,\cdot)\) \(\chi_{6900}(2593,\cdot)\) \(\chi_{6900}(2857,\cdot)\) \(\chi_{6900}(3193,\cdot)\) \(\chi_{6900}(3457,\cdot)\) \(\chi_{6900}(3493,\cdot)\) \(\chi_{6900}(3793,\cdot)\) \(\chi_{6900}(4357,\cdot)\) \(\chi_{6900}(4657,\cdot)\) \(\chi_{6900}(5557,\cdot)\) \(\chi_{6900}(5857,\cdot)\) \(\chi_{6900}(5893,\cdot)\) \(\chi_{6900}(6457,\cdot)\) \(\chi_{6900}(6493,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((3451,4601,277,1201)\) → \((1,1,i,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6900 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) |
sage: chi.jacobi_sum(n)