from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4410, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,21,10]))
pari: [g,chi] = znchar(Mod(2209,4410))
Basic properties
Modulus: | \(4410\) | |
Conductor: | \(2205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2205}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4410.cy
\(\chi_{4410}(319,\cdot)\) \(\chi_{4410}(709,\cdot)\) \(\chi_{4410}(1339,\cdot)\) \(\chi_{4410}(1579,\cdot)\) \(\chi_{4410}(1969,\cdot)\) \(\chi_{4410}(2209,\cdot)\) \(\chi_{4410}(2599,\cdot)\) \(\chi_{4410}(2839,\cdot)\) \(\chi_{4410}(3229,\cdot)\) \(\chi_{4410}(3469,\cdot)\) \(\chi_{4410}(3859,\cdot)\) \(\chi_{4410}(4099,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((3431,2647,1081)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{5}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4410 }(2209, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) |
sage: chi.jacobi_sum(n)