from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,28]))
pari: [g,chi] = znchar(Mod(13,2415))
Basic properties
| Modulus: | \(2415\) | |
| Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{805}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.co
\(\chi_{2415}(13,\cdot)\) \(\chi_{2415}(118,\cdot)\) \(\chi_{2415}(202,\cdot)\) \(\chi_{2415}(223,\cdot)\) \(\chi_{2415}(307,\cdot)\) \(\chi_{2415}(328,\cdot)\) \(\chi_{2415}(538,\cdot)\) \(\chi_{2415}(748,\cdot)\) \(\chi_{2415}(832,\cdot)\) \(\chi_{2415}(853,\cdot)\) \(\chi_{2415}(1168,\cdot)\) \(\chi_{2415}(1273,\cdot)\) \(\chi_{2415}(1462,\cdot)\) \(\chi_{2415}(1567,\cdot)\) \(\chi_{2415}(1672,\cdot)\) \(\chi_{2415}(1777,\cdot)\) \(\chi_{2415}(1798,\cdot)\) \(\chi_{2415}(1987,\cdot)\) \(\chi_{2415}(2197,\cdot)\) \(\chi_{2415}(2302,\cdot)\)
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
\((806,967,346,1891)\) → \((1,-i,-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(-i\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)