from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2760, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,11,2]))
pari: [g,chi] = znchar(Mod(97,2760))
Basic properties
Modulus: | \(2760\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2760.dm
\(\chi_{2760}(97,\cdot)\) \(\chi_{2760}(217,\cdot)\) \(\chi_{2760}(313,\cdot)\) \(\chi_{2760}(337,\cdot)\) \(\chi_{2760}(433,\cdot)\) \(\chi_{2760}(457,\cdot)\) \(\chi_{2760}(697,\cdot)\) \(\chi_{2760}(793,\cdot)\) \(\chi_{2760}(937,\cdot)\) \(\chi_{2760}(1033,\cdot)\) \(\chi_{2760}(1417,\cdot)\) \(\chi_{2760}(1537,\cdot)\) \(\chi_{2760}(1753,\cdot)\) \(\chi_{2760}(1873,\cdot)\) \(\chi_{2760}(1897,\cdot)\) \(\chi_{2760}(1993,\cdot)\) \(\chi_{2760}(2113,\cdot)\) \(\chi_{2760}(2137,\cdot)\) \(\chi_{2760}(2353,\cdot)\) \(\chi_{2760}(2593,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((2071,1381,1841,1657,1201)\) → \((1,1,1,i,e\left(\frac{1}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2760 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) |
sage: chi.jacobi_sum(n)