from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,0,29]))
pari: [g,chi] = znchar(Mod(6571,8820))
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{196}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8820.gt
\(\chi_{8820}(271,\cdot)\) \(\chi_{8820}(451,\cdot)\) \(\chi_{8820}(1531,\cdot)\) \(\chi_{8820}(1711,\cdot)\) \(\chi_{8820}(2791,\cdot)\) \(\chi_{8820}(4051,\cdot)\) \(\chi_{8820}(4231,\cdot)\) \(\chi_{8820}(5491,\cdot)\) \(\chi_{8820}(6571,\cdot)\) \(\chi_{8820}(6751,\cdot)\) \(\chi_{8820}(7831,\cdot)\) \(\chi_{8820}(8011,\cdot)\)
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: | \(\Q(\zeta_{196})^+\) |
Values on generators
\((4411,7841,7057,1081)\) → \((-1,1,1,e\left(\frac{29}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(6571, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)