from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,26]))
pari: [g,chi] = znchar(Mod(297,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{920}(757,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cm
\(\chi_{1840}(57,\cdot)\) \(\chi_{1840}(153,\cdot)\) \(\chi_{1840}(217,\cdot)\) \(\chi_{1840}(297,\cdot)\) \(\chi_{1840}(313,\cdot)\) \(\chi_{1840}(457,\cdot)\) \(\chi_{1840}(617,\cdot)\) \(\chi_{1840}(697,\cdot)\) \(\chi_{1840}(793,\cdot)\) \(\chi_{1840}(937,\cdot)\) \(\chi_{1840}(953,\cdot)\) \(\chi_{1840}(1017,\cdot)\) \(\chi_{1840}(1033,\cdot)\) \(\chi_{1840}(1193,\cdot)\) \(\chi_{1840}(1257,\cdot)\) \(\chi_{1840}(1353,\cdot)\) \(\chi_{1840}(1417,\cdot)\) \(\chi_{1840}(1433,\cdot)\) \(\chi_{1840}(1673,\cdot)\) \(\chi_{1840}(1753,\cdot)\)
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: | 44.44.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((1151,1381,737,1201)\) → \((1,-1,i,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(297, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)