from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4410, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,21,3]))
pari: [g,chi] = znchar(Mod(419,4410))
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 \(\chi_{2205}(419,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4410.dj
\(\chi_{4410}(209,\cdot)\) \(\chi_{4410}(419,\cdot)\) \(\chi_{4410}(839,\cdot)\) \(\chi_{4410}(1049,\cdot)\) \(\chi_{4410}(1679,\cdot)\) \(\chi_{4410}(2099,\cdot)\) \(\chi_{4410}(2309,\cdot)\) \(\chi_{4410}(2729,\cdot)\) \(\chi_{4410}(3359,\cdot)\) \(\chi_{4410}(3569,\cdot)\) \(\chi_{4410}(3989,\cdot)\) \(\chi_{4410}(4199,\cdot)\)
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: | 42.42.1316321998912547656676347346780699171487983838970489423466937863866231487028326848486202559309482574462890625.1 |
Values on generators
\((3431,2647,1081)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{1}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4410 }(419, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) |
sage: chi.jacobi_sum(n)