from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,22,38]))
pari: [g,chi] = znchar(Mod(1525,7728))
Basic properties
Modulus: | \(7728\) | |
Conductor: | \(2576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2576}(1525,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.fd
\(\chi_{7728}(181,\cdot)\) \(\chi_{7728}(517,\cdot)\) \(\chi_{7728}(1525,\cdot)\) \(\chi_{7728}(1693,\cdot)\) \(\chi_{7728}(1861,\cdot)\) \(\chi_{7728}(2029,\cdot)\) \(\chi_{7728}(2365,\cdot)\) \(\chi_{7728}(2701,\cdot)\) \(\chi_{7728}(2869,\cdot)\) \(\chi_{7728}(3373,\cdot)\) \(\chi_{7728}(4045,\cdot)\) \(\chi_{7728}(4381,\cdot)\) \(\chi_{7728}(5389,\cdot)\) \(\chi_{7728}(5557,\cdot)\) \(\chi_{7728}(5725,\cdot)\) \(\chi_{7728}(5893,\cdot)\) \(\chi_{7728}(6229,\cdot)\) \(\chi_{7728}(6565,\cdot)\) \(\chi_{7728}(6733,\cdot)\) \(\chi_{7728}(7237,\cdot)\)
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
\((4831,5797,5153,6625,6721)\) → \((1,i,1,-1,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 7728 }(1525, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)