from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4410, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,0,9]))
pari: [g,chi] = znchar(Mod(3611,4410))
Basic properties
Modulus: | \(4410\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4410.dd
\(\chi_{4410}(41,\cdot)\) \(\chi_{4410}(461,\cdot)\) \(\chi_{4410}(671,\cdot)\) \(\chi_{4410}(1091,\cdot)\) \(\chi_{4410}(1301,\cdot)\) \(\chi_{4410}(1721,\cdot)\) \(\chi_{4410}(1931,\cdot)\) \(\chi_{4410}(2561,\cdot)\) \(\chi_{4410}(2981,\cdot)\) \(\chi_{4410}(3191,\cdot)\) \(\chi_{4410}(3611,\cdot)\) \(\chi_{4410}(4241,\cdot)\)
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
\((3431,2647,1081)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4410 }(3611, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)