from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,33,10]))
pari: [g,chi] = znchar(Mod(3493,5520))
Basic properties
Modulus: | \(5520\) | |
Conductor: | \(1840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1840}(1653,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.fb
\(\chi_{5520}(157,\cdot)\) \(\chi_{5520}(373,\cdot)\) \(\chi_{5520}(613,\cdot)\) \(\chi_{5520}(1597,\cdot)\) \(\chi_{5520}(1813,\cdot)\) \(\chi_{5520}(1837,\cdot)\) \(\chi_{5520}(2077,\cdot)\) \(\chi_{5520}(2317,\cdot)\) \(\chi_{5520}(2797,\cdot)\) \(\chi_{5520}(3253,\cdot)\) \(\chi_{5520}(3277,\cdot)\) \(\chi_{5520}(3493,\cdot)\) \(\chi_{5520}(3517,\cdot)\) \(\chi_{5520}(3733,\cdot)\) \(\chi_{5520}(3973,\cdot)\) \(\chi_{5520}(4237,\cdot)\) \(\chi_{5520}(4453,\cdot)\) \(\chi_{5520}(4477,\cdot)\) \(\chi_{5520}(4933,\cdot)\) \(\chi_{5520}(5173,\cdot)\)
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
\((4831,1381,1841,4417,1201)\) → \((1,i,1,-i,e\left(\frac{5}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5520 }(3493, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)