from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,11,12]))
pari: [g,chi] = znchar(Mod(77,5520))
Basic properties
Modulus: | \(5520\) | |
Conductor: | \(5520\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.ey
\(\chi_{5520}(77,\cdot)\) \(\chi_{5520}(317,\cdot)\) \(\chi_{5520}(533,\cdot)\) \(\chi_{5520}(1037,\cdot)\) \(\chi_{5520}(1277,\cdot)\) \(\chi_{5520}(1733,\cdot)\) \(\chi_{5520}(1757,\cdot)\) \(\chi_{5520}(1973,\cdot)\) \(\chi_{5520}(2237,\cdot)\) \(\chi_{5520}(2477,\cdot)\) \(\chi_{5520}(2693,\cdot)\) \(\chi_{5520}(2717,\cdot)\) \(\chi_{5520}(2933,\cdot)\) \(\chi_{5520}(2957,\cdot)\) \(\chi_{5520}(3413,\cdot)\) \(\chi_{5520}(3893,\cdot)\) \(\chi_{5520}(4133,\cdot)\) \(\chi_{5520}(4373,\cdot)\) \(\chi_{5520}(4397,\cdot)\) \(\chi_{5520}(4613,\cdot)\)
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
\((4831,1381,1841,4417,1201)\) → \((1,-i,-1,i,e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5520 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)