from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,11,20]))
pari: [g,chi] = znchar(Mod(607,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{460}(147,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cr
\(\chi_{1840}(127,\cdot)\) \(\chi_{1840}(223,\cdot)\) \(\chi_{1840}(303,\cdot)\) \(\chi_{1840}(463,\cdot)\) \(\chi_{1840}(607,\cdot)\) \(\chi_{1840}(623,\cdot)\) \(\chi_{1840}(703,\cdot)\) \(\chi_{1840}(767,\cdot)\) \(\chi_{1840}(863,\cdot)\) \(\chi_{1840}(1007,\cdot)\) \(\chi_{1840}(1087,\cdot)\) \(\chi_{1840}(1327,\cdot)\) \(\chi_{1840}(1343,\cdot)\) \(\chi_{1840}(1407,\cdot)\) \(\chi_{1840}(1503,\cdot)\) \(\chi_{1840}(1567,\cdot)\) \(\chi_{1840}(1727,\cdot)\) \(\chi_{1840}(1743,\cdot)\) \(\chi_{1840}(1807,\cdot)\) \(\chi_{1840}(1823,\cdot)\)
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
\((1151,1381,737,1201)\) → \((-1,1,i,e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(607, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)