from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,33,28]))
pari: [g,chi] = znchar(Mod(13,1840))
Basic properties
| Modulus: | \(1840\) | |
| Conductor: | \(1840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.db
\(\chi_{1840}(13,\cdot)\) \(\chi_{1840}(117,\cdot)\) \(\chi_{1840}(173,\cdot)\) \(\chi_{1840}(197,\cdot)\) \(\chi_{1840}(357,\cdot)\) \(\chi_{1840}(653,\cdot)\) \(\chi_{1840}(813,\cdot)\) \(\chi_{1840}(837,\cdot)\) \(\chi_{1840}(997,\cdot)\) \(\chi_{1840}(1053,\cdot)\) \(\chi_{1840}(1133,\cdot)\) \(\chi_{1840}(1237,\cdot)\) \(\chi_{1840}(1317,\cdot)\) \(\chi_{1840}(1373,\cdot)\) \(\chi_{1840}(1453,\cdot)\) \(\chi_{1840}(1557,\cdot)\) \(\chi_{1840}(1613,\cdot)\) \(\chi_{1840}(1637,\cdot)\) \(\chi_{1840}(1773,\cdot)\) \(\chi_{1840}(1797,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((1151,1381,737,1201)\) → \((1,-i,-i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) |
sage: chi.jacobi_sum(n)