from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2592, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,17]))
pari: [g,chi] = znchar(Mod(95,2592))
Basic properties
| Modulus: | \(2592\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(95,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2592.by
\(\chi_{2592}(95,\cdot)\) \(\chi_{2592}(191,\cdot)\) \(\chi_{2592}(383,\cdot)\) \(\chi_{2592}(479,\cdot)\) \(\chi_{2592}(671,\cdot)\) \(\chi_{2592}(767,\cdot)\) \(\chi_{2592}(959,\cdot)\) \(\chi_{2592}(1055,\cdot)\) \(\chi_{2592}(1247,\cdot)\) \(\chi_{2592}(1343,\cdot)\) \(\chi_{2592}(1535,\cdot)\) \(\chi_{2592}(1631,\cdot)\) \(\chi_{2592}(1823,\cdot)\) \(\chi_{2592}(1919,\cdot)\) \(\chi_{2592}(2111,\cdot)\) \(\chi_{2592}(2207,\cdot)\) \(\chi_{2592}(2399,\cdot)\) \(\chi_{2592}(2495,\cdot)\)
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
\((2431,325,1217)\) → \((-1,1,e\left(\frac{17}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2592 }(95, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{43}{54}\right)\) |
sage: chi.jacobi_sum(n)