from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7938, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([31,9]))
pari: [g,chi] = znchar(Mod(227,7938))
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 \(\chi_{567}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7938.ce
\(\chi_{7938}(227,\cdot)\) \(\chi_{7938}(509,\cdot)\) \(\chi_{7938}(1109,\cdot)\) \(\chi_{7938}(1391,\cdot)\) \(\chi_{7938}(1991,\cdot)\) \(\chi_{7938}(2273,\cdot)\) \(\chi_{7938}(2873,\cdot)\) \(\chi_{7938}(3155,\cdot)\) \(\chi_{7938}(3755,\cdot)\) \(\chi_{7938}(4037,\cdot)\) \(\chi_{7938}(4637,\cdot)\) \(\chi_{7938}(4919,\cdot)\) \(\chi_{7938}(5519,\cdot)\) \(\chi_{7938}(5801,\cdot)\) \(\chi_{7938}(6401,\cdot)\) \(\chi_{7938}(6683,\cdot)\) \(\chi_{7938}(7283,\cdot)\) \(\chi_{7938}(7565,\cdot)\)
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
\((6077,3727)\) → \((e\left(\frac{31}{54}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7938 }(227, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)