from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,42]))
pari: [g,chi] = znchar(Mod(83,2415))
Basic properties
| Modulus: | \(2415\) | |
| Conductor: | \(2415\) | 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 2415.cp
\(\chi_{2415}(83,\cdot)\) \(\chi_{2415}(272,\cdot)\) \(\chi_{2415}(293,\cdot)\) \(\chi_{2415}(398,\cdot)\) \(\chi_{2415}(503,\cdot)\) \(\chi_{2415}(608,\cdot)\) \(\chi_{2415}(797,\cdot)\) \(\chi_{2415}(902,\cdot)\) \(\chi_{2415}(1217,\cdot)\) \(\chi_{2415}(1238,\cdot)\) \(\chi_{2415}(1322,\cdot)\) \(\chi_{2415}(1532,\cdot)\) \(\chi_{2415}(1742,\cdot)\) \(\chi_{2415}(1763,\cdot)\) \(\chi_{2415}(1847,\cdot)\) \(\chi_{2415}(1868,\cdot)\) \(\chi_{2415}(1952,\cdot)\) \(\chi_{2415}(2057,\cdot)\) \(\chi_{2415}(2183,\cdot)\) \(\chi_{2415}(2288,\cdot)\)
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
\((806,967,346,1891)\) → \((-1,-i,-1,e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(i\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)