from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7220, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,0,1]))
pari: [g,chi] = znchar(Mod(151,7220))
Basic properties
Modulus: | \(7220\) | |
Conductor: | \(1444\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1444}(151,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7220.bn
\(\chi_{7220}(151,\cdot)\) \(\chi_{7220}(531,\cdot)\) \(\chi_{7220}(911,\cdot)\) \(\chi_{7220}(1291,\cdot)\) \(\chi_{7220}(1671,\cdot)\) \(\chi_{7220}(2051,\cdot)\) \(\chi_{7220}(2431,\cdot)\) \(\chi_{7220}(2811,\cdot)\) \(\chi_{7220}(3191,\cdot)\) \(\chi_{7220}(3571,\cdot)\) \(\chi_{7220}(3951,\cdot)\) \(\chi_{7220}(4711,\cdot)\) \(\chi_{7220}(5091,\cdot)\) \(\chi_{7220}(5471,\cdot)\) \(\chi_{7220}(5851,\cdot)\) \(\chi_{7220}(6231,\cdot)\) \(\chi_{7220}(6611,\cdot)\) \(\chi_{7220}(6991,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | 38.38.613977851508884585934104054696506599029285341338287807495000891446051265879187437942845703843204086890496.1 |
Values on generators
\((3611,5777,6861)\) → \((-1,1,e\left(\frac{1}{38}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 7220 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) |
sage: chi.jacobi_sum(n)