from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,0,30]))
pari: [g,chi] = znchar(Mod(3331,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{368}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.gc
\(\chi_{5520}(451,\cdot)\) \(\chi_{5520}(571,\cdot)\) \(\chi_{5520}(931,\cdot)\) \(\chi_{5520}(1171,\cdot)\) \(\chi_{5520}(1891,\cdot)\) \(\chi_{5520}(2011,\cdot)\) \(\chi_{5520}(2131,\cdot)\) \(\chi_{5520}(2251,\cdot)\) \(\chi_{5520}(2491,\cdot)\) \(\chi_{5520}(2731,\cdot)\) \(\chi_{5520}(3211,\cdot)\) \(\chi_{5520}(3331,\cdot)\) \(\chi_{5520}(3691,\cdot)\) \(\chi_{5520}(3931,\cdot)\) \(\chi_{5520}(4651,\cdot)\) \(\chi_{5520}(4771,\cdot)\) \(\chi_{5520}(4891,\cdot)\) \(\chi_{5520}(5011,\cdot)\) \(\chi_{5520}(5251,\cdot)\) \(\chi_{5520}(5491,\cdot)\)
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: | 44.44.4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608.1 |
Values on generators
\((4831,1381,1841,4417,1201)\) → \((-1,-i,1,1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(3331, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)