from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,7]))
pari: [g,chi] = znchar(Mod(109,4005))
Basic properties
| Modulus: | \(4005\) | |
| Conductor: | \(445\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{445}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4005.cx
\(\chi_{4005}(109,\cdot)\) \(\chi_{4005}(199,\cdot)\) \(\chi_{4005}(424,\cdot)\) \(\chi_{4005}(514,\cdot)\) \(\chi_{4005}(694,\cdot)\) \(\chi_{4005}(784,\cdot)\) \(\chi_{4005}(1414,\cdot)\) \(\chi_{4005}(1504,\cdot)\) \(\chi_{4005}(1549,\cdot)\) \(\chi_{4005}(1864,\cdot)\) \(\chi_{4005}(1909,\cdot)\) \(\chi_{4005}(2089,\cdot)\) \(\chi_{4005}(2539,\cdot)\) \(\chi_{4005}(2719,\cdot)\) \(\chi_{4005}(2764,\cdot)\) \(\chi_{4005}(3079,\cdot)\) \(\chi_{4005}(3124,\cdot)\) \(\chi_{4005}(3214,\cdot)\) \(\chi_{4005}(3844,\cdot)\) \(\chi_{4005}(3934,\cdot)\)
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: | 44.44.1588950548018177733817067321623614819860239312933539604872830957860279621432150312826633453369140625.1 |
Values on generators
\((3116,802,181)\) → \((1,-1,e\left(\frac{7}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(109, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) |
sage: chi.jacobi_sum(n)