from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,34]))
pari: [g,chi] = znchar(Mod(1853,2736))
Basic properties
| Modulus: | \(2736\) | |
| Conductor: | \(912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{912}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2736.hc
\(\chi_{2736}(53,\cdot)\) \(\chi_{2736}(269,\cdot)\) \(\chi_{2736}(413,\cdot)\) \(\chi_{2736}(485,\cdot)\) \(\chi_{2736}(629,\cdot)\) \(\chi_{2736}(773,\cdot)\) \(\chi_{2736}(1421,\cdot)\) \(\chi_{2736}(1637,\cdot)\) \(\chi_{2736}(1781,\cdot)\) \(\chi_{2736}(1853,\cdot)\) \(\chi_{2736}(1997,\cdot)\) \(\chi_{2736}(2141,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.7375213349858562030923708432825799203687776914247215677306393587785801919440617472.1 |
Values on generators
\((1711,2053,1217,1009)\) → \((1,-i,-1,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2736 }(1853, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) |
sage: chi.jacobi_sum(n)