from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,38]))
pari: [g,chi] = znchar(Mod(513,7360))
Basic properties
Modulus: | \(7360\) | |
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 \(\chi_{115}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.dz
\(\chi_{7360}(513,\cdot)\) \(\chi_{7360}(833,\cdot)\) \(\chi_{7360}(1217,\cdot)\) \(\chi_{7360}(1537,\cdot)\) \(\chi_{7360}(1857,\cdot)\) \(\chi_{7360}(2113,\cdot)\) \(\chi_{7360}(2177,\cdot)\) \(\chi_{7360}(2817,\cdot)\) \(\chi_{7360}(3073,\cdot)\) \(\chi_{7360}(3457,\cdot)\) \(\chi_{7360}(3713,\cdot)\) \(\chi_{7360}(3777,\cdot)\) \(\chi_{7360}(5057,\cdot)\) \(\chi_{7360}(5633,\cdot)\) \(\chi_{7360}(5953,\cdot)\) \(\chi_{7360}(6017,\cdot)\) \(\chi_{7360}(6273,\cdot)\) \(\chi_{7360}(6593,\cdot)\) \(\chi_{7360}(6657,\cdot)\) \(\chi_{7360}(7233,\cdot)\)
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
\((1151,5061,4417,6721)\) → \((1,1,-i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 7360 }(513, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)