from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2760, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,33,32]))
pari: [g,chi] = znchar(Mod(233,2760))
Basic properties
Modulus: | \(2760\) | |
Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{345}(233,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2760.do
\(\chi_{2760}(233,\cdot)\) \(\chi_{2760}(257,\cdot)\) \(\chi_{2760}(353,\cdot)\) \(\chi_{2760}(377,\cdot)\) \(\chi_{2760}(473,\cdot)\) \(\chi_{2760}(593,\cdot)\) \(\chi_{2760}(857,\cdot)\) \(\chi_{2760}(1097,\cdot)\) \(\chi_{2760}(1313,\cdot)\) \(\chi_{2760}(1337,\cdot)\) \(\chi_{2760}(1457,\cdot)\) \(\chi_{2760}(1553,\cdot)\) \(\chi_{2760}(1577,\cdot)\) \(\chi_{2760}(1697,\cdot)\) \(\chi_{2760}(1913,\cdot)\) \(\chi_{2760}(2033,\cdot)\) \(\chi_{2760}(2417,\cdot)\) \(\chi_{2760}(2513,\cdot)\) \(\chi_{2760}(2657,\cdot)\) \(\chi_{2760}(2753,\cdot)\)
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
\((2071,1381,1841,1657,1201)\) → \((1,1,-1,-i,e\left(\frac{8}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2760 }(233, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) |
sage: chi.jacobi_sum(n)