from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,22,11,10]))
pari: [g,chi] = znchar(Mod(407,8040))
Basic properties
Modulus: | \(8040\) | |
Conductor: | \(4020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4020}(407,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.fk
\(\chi_{8040}(407,\cdot)\) \(\chi_{8040}(527,\cdot)\) \(\chi_{8040}(983,\cdot)\) \(\chi_{8040}(1343,\cdot)\) \(\chi_{8040}(1367,\cdot)\) \(\chi_{8040}(1583,\cdot)\) \(\chi_{8040}(1727,\cdot)\) \(\chi_{8040}(2063,\cdot)\) \(\chi_{8040}(3023,\cdot)\) \(\chi_{8040}(3527,\cdot)\) \(\chi_{8040}(3623,\cdot)\) \(\chi_{8040}(3743,\cdot)\) \(\chi_{8040}(4583,\cdot)\) \(\chi_{8040}(4943,\cdot)\) \(\chi_{8040}(5807,\cdot)\) \(\chi_{8040}(6167,\cdot)\) \(\chi_{8040}(6407,\cdot)\) \(\chi_{8040}(6743,\cdot)\) \(\chi_{8040}(6887,\cdot)\) \(\chi_{8040}(7847,\cdot)\)
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
\((6031,4021,2681,3217,5161)\) → \((-1,1,-1,i,e\left(\frac{5}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 8040 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)