from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,0,10]))
pari: [g,chi] = znchar(Mod(365,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(1104\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1104}(365,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.ez
\(\chi_{7728}(365,\cdot)\) \(\chi_{7728}(701,\cdot)\) \(\chi_{7728}(1709,\cdot)\) \(\chi_{7728}(1877,\cdot)\) \(\chi_{7728}(2045,\cdot)\) \(\chi_{7728}(2213,\cdot)\) \(\chi_{7728}(2549,\cdot)\) \(\chi_{7728}(2885,\cdot)\) \(\chi_{7728}(3053,\cdot)\) \(\chi_{7728}(3557,\cdot)\) \(\chi_{7728}(4229,\cdot)\) \(\chi_{7728}(4565,\cdot)\) \(\chi_{7728}(5573,\cdot)\) \(\chi_{7728}(5741,\cdot)\) \(\chi_{7728}(5909,\cdot)\) \(\chi_{7728}(6077,\cdot)\) \(\chi_{7728}(6413,\cdot)\) \(\chi_{7728}(6749,\cdot)\) \(\chi_{7728}(6917,\cdot)\) \(\chi_{7728}(7421,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((1,-i,-1,1,e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(365, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)