from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,42]))
pari: [g,chi] = znchar(Mod(37,2070))
Basic properties
Modulus: | \(2070\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2070.bj
\(\chi_{2070}(37,\cdot)\) \(\chi_{2070}(217,\cdot)\) \(\chi_{2070}(343,\cdot)\) \(\chi_{2070}(433,\cdot)\) \(\chi_{2070}(523,\cdot)\) \(\chi_{2070}(613,\cdot)\) \(\chi_{2070}(757,\cdot)\) \(\chi_{2070}(793,\cdot)\) \(\chi_{2070}(847,\cdot)\) \(\chi_{2070}(937,\cdot)\) \(\chi_{2070}(973,\cdot)\) \(\chi_{2070}(1027,\cdot)\) \(\chi_{2070}(1063,\cdot)\) \(\chi_{2070}(1207,\cdot)\) \(\chi_{2070}(1387,\cdot)\) \(\chi_{2070}(1423,\cdot)\) \(\chi_{2070}(1477,\cdot)\) \(\chi_{2070}(1693,\cdot)\) \(\chi_{2070}(1837,\cdot)\) \(\chi_{2070}(1873,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((461,1657,1891)\) → \((1,i,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2070 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) |
sage: chi.jacobi_sum(n)