from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7938, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([47,27]))
pari: [g,chi] = znchar(Mod(293,7938))
Basic properties
| Modulus: | \(7938\) | |
| Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{567}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7938.ck
\(\chi_{7938}(293,\cdot)\) \(\chi_{7938}(587,\cdot)\) \(\chi_{7938}(1175,\cdot)\) \(\chi_{7938}(1469,\cdot)\) \(\chi_{7938}(2057,\cdot)\) \(\chi_{7938}(2351,\cdot)\) \(\chi_{7938}(2939,\cdot)\) \(\chi_{7938}(3233,\cdot)\) \(\chi_{7938}(3821,\cdot)\) \(\chi_{7938}(4115,\cdot)\) \(\chi_{7938}(4703,\cdot)\) \(\chi_{7938}(4997,\cdot)\) \(\chi_{7938}(5585,\cdot)\) \(\chi_{7938}(5879,\cdot)\) \(\chi_{7938}(6467,\cdot)\) \(\chi_{7938}(6761,\cdot)\) \(\chi_{7938}(7349,\cdot)\) \(\chi_{7938}(7643,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((6077,3727)\) → \((e\left(\frac{47}{54}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7938 }(293, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)