from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1104, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,30]))
pari: [g,chi] = znchar(Mod(19,1104))
Basic properties
| Modulus: | \(1104\) | |
| 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 1104.bv
\(\chi_{1104}(19,\cdot)\) \(\chi_{1104}(43,\cdot)\) \(\chi_{1104}(67,\cdot)\) \(\chi_{1104}(235,\cdot)\) \(\chi_{1104}(283,\cdot)\) \(\chi_{1104}(355,\cdot)\) \(\chi_{1104}(379,\cdot)\) \(\chi_{1104}(451,\cdot)\) \(\chi_{1104}(475,\cdot)\) \(\chi_{1104}(523,\cdot)\) \(\chi_{1104}(571,\cdot)\) \(\chi_{1104}(595,\cdot)\) \(\chi_{1104}(619,\cdot)\) \(\chi_{1104}(787,\cdot)\) \(\chi_{1104}(835,\cdot)\) \(\chi_{1104}(907,\cdot)\) \(\chi_{1104}(931,\cdot)\) \(\chi_{1104}(1003,\cdot)\) \(\chi_{1104}(1027,\cdot)\) \(\chi_{1104}(1075,\cdot)\)
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
\((415,277,737,97)\) → \((-1,-i,1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1104 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) |
sage: chi.jacobi_sum(n)